The actual code is very big, I tried to simplify it as much as possible.
The design is such we cannot call the continuation method directly. So looking for ways of invoking it.
VF page



Controller:



Is it True , Continuation callback is invoked only when a continuation is initialized from Visualforce only. It will not work if continuation is initiated by any Apex.If yes, then how can we use Continuation in this scenario. Many Thanks

Component:
<aura:component controller="RemovefromDNCcontroller"
                implements="force:appHostable,flexipage:availableForAllPageTypes,flexipage:availableForRecordHome,force:hasRecordId,force:lightningQuickActionWithoutHeader" access="global">
    <ltng:require styles="{!$Resource.quickActionStyle}"/>
    <aura:attribute name="recordId" type="Id" />
    <aura:attribute name="Preval" type="RemovefromDNCcontroller" />
    <aura:attribute name="Preaccount" type="Id"/>
    <aura:attribute name="Precontact" type="Id"/>
    <aura:handler name="init" value="{!this}" action="{!c.doInit}" />
    <aura:if isTrue = "{!v.Preval.messageDisplay}">
        <div class="slds-notify slds-notify_alert slds-theme_alert-texture slds-theme_info" role="alert">
            {!v.Preval.message}
        </div>
    </aura:if>
</aura:component>

Controller.js

({
    doInit : function(component, event,helper) {
        var action = component.get("c.Oppdetails");
        action.setParams({"OppId": component.get("v.recordId")});
        action.setCallback(this, function(response) {
            var responsedata = response.getReturnValue();
            var accid = response.getReturnValue().caseaccId;
            var conid = response.getReturnValue().caseContactId;
            var state = response.getState();
            console.log('accid==='+accid);
            console.log('conid==='+conid);
            helper.fetchAccdetail(component,event,helper,accid);
            helper.fetchCondetails(component, event, helper,conid);
            if (component.isValid() && state === "SUCCESS") {
                component.set("v.Preval", response.getReturnValue());
                helper.createCasDetails(component,event,helper,responsedata);
            }
        });
        $A.enqueueAction(action);
    },
})

Helper.js
({
    fetchAccdetail : function(component, event, helper,accid) {
        console.log('accid helper==='+accid);
        var action = component.get("c.accountPreCheck");
        action.setParams({
            "AccountcheckId" : accid
        });
        
        action.setCallback(this, function(response){
            var state = response.getState();
            console.log('response acc===='+response.getReturnValue());            
            if (state === "SUCCESS") {
                component.set("v.Preaccount", response.getReturnValue());
                //var Acc = component.get("v.Preaccount");
                //console.log("Acc---"+Acc);
            }
        });
        $A.enqueueAction(action);
    },
    fetchCondetails : function(component, event, helper,conid) {
        console.log('conid helper==='+conid);
        var action = component.get("c.contactPreCheck");
        action.setParams({
            "ContactcheckId" : conid
        });
        
        action.setCallback(this, function(response){
            var state = response.getState();
            console.log('response con==='+response.getReturnValue());            
            if (state === "SUCCESS") {
                component.set("v.Precontact", response.getReturnValue());
                //var con = component.get("v.Precontact");
                //console.log("con---"+con);
            }
        });
        $A.enqueueAction(action);
    },
    createCasDetails: function(component,event,helper,responsedata){
        console.log("responsedata---"+responsedata.messageDisplay);
        if(responsedata.messageDisplay == false){
            var Acc = component.get("v.Preaccount");
            var con = component.get("v.Precontact");
            console.log("Acccase---"+Acc);
            console.log("concase---"+con);
            var createCaseEvent = $A.get("e.force:createRecord");
            createCaseEvent.setParams({
                "entityApiName": "Case",
                "defaultFieldValues": {
                    'Case_Type__c'     : 'Do Not Call List Request',
                    'Account'        : Acc,
                    'Contact'        : con,
                    'Opportunity__c': responsedata.caseOppId
                }
            });
            createCaseEvent.fire();
        }
    },
})

Apex Class:

public with sharing class RemovefromDNCcontroller {

    @AuraEnabled public string caseaccId{get;set;}
    @AuraEnabled public string caseContactId{get;set;}
    @AuraEnabled public string caseOppId{get;set;}
    @AuraEnabled public string message{get;set;}
    @AuraEnabled public boolean messageDisplay{get;set;}
    
    @AuraEnabled 
    public static RemovefromDNCcontroller Oppdetails (Id OppId){
        RemovefromDNCcontroller rfd = new RemovefromDNCcontroller();
         rfd.messageDisplay = false;
        Opportunity o = [select id, Name,ownerId, AccountId, Account.PersonContactId, Do_Not_Call__c from Opportunity where Id =: OppId];
        system.debug('o----'+o);
        if(o.Do_Not_Call__c == true){
            system.debug('rfd 24'+ rfd.message);
            rfd.caseOppId = o.Id;
            rfd.caseContactId = o.Account.PersonContactId;
            rfd.caseaccId = o.AccountId;
            system.debug('caseContactId ==='+rfd.caseContactId);
            system.debug('caseaccId ==='+rfd.caseaccId);
            return rfd;
        }else{
            rfd.message = system.Label.Remove_from_DNC_Alert;
            rfd.messageDisplay = true;
            system.debug('rfd 32'+ rfd.message);
            return rfd;
            
        }
        
    }
    @AuraEnabled 
    public static Id accountPreCheck (Id AccountcheckId){
        system.debug('AccountcheckId--'+AccountcheckId);
        return AccountcheckId;
    }
    @AuraEnabled 
    public static Id contactPreCheck (Id ContactcheckId){
        system.debug('ContactcheckId--'+ContactcheckId);
        return ContactcheckId;
    }
}

In Helper, method createCasDetails, values in 
console.log("Acccase---"+Acc);
console.log("concase---"+con);
are not coming due to which values in Account and Contact are not pre-populating.
Please help.

Component:
<aura:component controller="RemovefromDNCcontroller"
                implements="force:appHostable,flexipage:availableForAllPageTypes,flexipage:availableForRecordHome,force:hasRecordId,force:lightningQuickActionWithoutHeader" access="global">
    <ltng:require styles="{!$Resource.quickActionStyle}"/>
    <aura:attribute name="recordId" type="Id" />
    <aura:attribute name="Preval" type="RemovefromDNCcontroller" />
    <aura:attribute name="Preaccount" type="Id"/>
    <aura:attribute name="Precontact" type="Id"/>
    <aura:handler name="init" value="{!this}" action="{!c.doInit}" />
    <aura:if isTrue = "{!v.Preval.messageDisplay}">
        <div class="slds-notify slds-notify_alert slds-theme_alert-texture slds-theme_info" role="alert">
            {!v.Preval.message}
        </div>
    </aura:if>
</aura:component>

Controller.js

({
    doInit : function(component, event,helper) {
        var action = component.get("c.Oppdetails");
        action.setParams({"OppId": component.get("v.recordId")});
        action.setCallback(this, function(response) {
            var responsedata = response.getReturnValue();
            var accid = response.getReturnValue().caseaccId;
            var conid = response.getReturnValue().caseContactId;
            var state = response.getState();
            console.log('accid==='+accid);
            console.log('conid==='+conid);
            helper.fetchAccdetail(component,event,helper,accid);
            helper.fetchCondetails(component, event, helper,conid);
            if (component.isValid() && state === "SUCCESS") {
                component.set("v.Preval", response.getReturnValue());
                helper.createCasDetails(component,event,helper,responsedata);
            }
        });
        $A.enqueueAction(action);
    },
})

Helper.js
({
    fetchAccdetail : function(component, event, helper,accid) {
        console.log('accid helper==='+accid);
        var action = component.get("c.accountPreCheck");
        action.setParams({
            "AccountcheckId" : accid
        });
        
        action.setCallback(this, function(response){
            var state = response.getState();
            console.log('response acc===='+response.getReturnValue());            
            if (state === "SUCCESS") {
                component.set("v.Preaccount", response.getReturnValue());
                //var Acc = component.get("v.Preaccount");
                //console.log("Acc---"+Acc);
            }
        });
        $A.enqueueAction(action);
    },
    fetchCondetails : function(component, event, helper,conid) {
        console.log('conid helper==='+conid);
        var action = component.get("c.contactPreCheck");
        action.setParams({
            "ContactcheckId" : conid
        });
        
        action.setCallback(this, function(response){
            var state = response.getState();
            console.log('response con==='+response.getReturnValue());            
            if (state === "SUCCESS") {
                component.set("v.Precontact", response.getReturnValue());
                //var con = component.get("v.Precontact");
                //console.log("con---"+con);
            }
        });
        $A.enqueueAction(action);
    },
    createCasDetails: function(component,event,helper,responsedata){
        console.log("responsedata---"+responsedata.messageDisplay);
        if(responsedata.messageDisplay == false){
            var Acc = component.get("v.Preaccount");
            var con = component.get("v.Precontact");
            console.log("Acccase---"+Acc);
            console.log("concase---"+con);
            var createCaseEvent = $A.get("e.force:createRecord");
            createCaseEvent.setParams({
                "entityApiName": "Case",
                "defaultFieldValues": {
                    'Case_Type__c'     : 'Do Not Call List Request',
                    'Account'        : Acc,
                    'Contact'        : con,
                    'Opportunity__c': responsedata.caseOppId
                }
            });
            createCaseEvent.fire();
        }
    },
})

Apex Class:

public with sharing class RemovefromDNCcontroller {

    @AuraEnabled public string caseaccId{get;set;}
    @AuraEnabled public string caseContactId{get;set;}
    @AuraEnabled public string caseOppId{get;set;}
    @AuraEnabled public string message{get;set;}
    @AuraEnabled public boolean messageDisplay{get;set;}
    
    @AuraEnabled 
    public static RemovefromDNCcontroller Oppdetails (Id OppId){
        RemovefromDNCcontroller rfd = new RemovefromDNCcontroller();
         rfd.messageDisplay = false;
        Opportunity o = [select id, Name,ownerId, AccountId, Account.PersonContactId, Do_Not_Call__c from Opportunity where Id =: OppId];
        system.debug('o----'+o);
        if(o.Do_Not_Call__c == true){
            system.debug('rfd 24'+ rfd.message);
            rfd.caseOppId = o.Id;
            rfd.caseContactId = o.Account.PersonContactId;
            rfd.caseaccId = o.AccountId;
            system.debug('caseContactId ==='+rfd.caseContactId);
            system.debug('caseaccId ==='+rfd.caseaccId);
            return rfd;
        }else{
            rfd.message = system.Label.Remove_from_DNC_Alert;
            rfd.messageDisplay = true;
            system.debug('rfd 32'+ rfd.message);
            return rfd;
            
        }
        
    }
    @AuraEnabled 
    public static Id accountPreCheck (Id AccountcheckId){
        system.debug('AccountcheckId--'+AccountcheckId);
        return AccountcheckId;
    }
    @AuraEnabled 
    public static Id contactPreCheck (Id ContactcheckId){
        system.debug('ContactcheckId--'+ContactcheckId);
        return ContactcheckId;
    }
}

In Helper, method createCasDetails, values in 
console.log("Acccase---"+Acc);
console.log("concase---"+con);
are not coming due to which values in Account and Contact are not pre-populating.
Please help.

The actual code is very big, I tried to simplify it as much as possible.
The design is such we cannot call the continuation method directly. So looking for ways of invoking it.
VF page



Controller:



Is it True , Continuation callback is invoked only when a continuation is initialized from Visualforce only. It will not work if continuation is initiated by any Apex.If yes, then how can we use Continuation in this scenario. Many Thanks

Component:
<aura:component controller="RemovefromDNCcontroller"
                implements="force:appHostable,flexipage:availableForAllPageTypes,flexipage:availableForRecordHome,force:hasRecordId,force:lightningQuickActionWithoutHeader" access="global">
    <ltng:require styles="{!$Resource.quickActionStyle}"/>
    <aura:attribute name="recordId" type="Id" />
    <aura:attribute name="Preval" type="RemovefromDNCcontroller" />
    <aura:attribute name="Preaccount" type="Id"/>
    <aura:attribute name="Precontact" type="Id"/>
    <aura:handler name="init" value="{!this}" action="{!c.doInit}" />
    <aura:if isTrue = "{!v.Preval.messageDisplay}">
        <div class="slds-notify slds-notify_alert slds-theme_alert-texture slds-theme_info" role="alert">
            {!v.Preval.message}
        </div>
    </aura:if>
</aura:component>

Controller.js

({
    doInit : function(component, event,helper) {
        var action = component.get("c.Oppdetails");
        action.setParams({"OppId": component.get("v.recordId")});
        action.setCallback(this, function(response) {
            var responsedata = response.getReturnValue();
            var accid = response.getReturnValue().caseaccId;
            var conid = response.getReturnValue().caseContactId;
            var state = response.getState();
            console.log('accid==='+accid);
            console.log('conid==='+conid);
            helper.fetchAccdetail(component,event,helper,accid);
            helper.fetchCondetails(component, event, helper,conid);
            if (component.isValid() && state === "SUCCESS") {
                component.set("v.Preval", response.getReturnValue());
                helper.createCasDetails(component,event,helper,responsedata);
            }
        });
        $A.enqueueAction(action);
    },
})

Helper.js
({
    fetchAccdetail : function(component, event, helper,accid) {
        console.log('accid helper==='+accid);
        var action = component.get("c.accountPreCheck");
        action.setParams({
            "AccountcheckId" : accid
        });
        
        action.setCallback(this, function(response){
            var state = response.getState();
            console.log('response acc===='+response.getReturnValue());            
            if (state === "SUCCESS") {
                component.set("v.Preaccount", response.getReturnValue());
                //var Acc = component.get("v.Preaccount");
                //console.log("Acc---"+Acc);
            }
        });
        $A.enqueueAction(action);
    },
    fetchCondetails : function(component, event, helper,conid) {
        console.log('conid helper==='+conid);
        var action = component.get("c.contactPreCheck");
        action.setParams({
            "ContactcheckId" : conid
        });
        
        action.setCallback(this, function(response){
            var state = response.getState();
            console.log('response con==='+response.getReturnValue());            
            if (state === "SUCCESS") {
                component.set("v.Precontact", response.getReturnValue());
                //var con = component.get("v.Precontact");
                //console.log("con---"+con);
            }
        });
        $A.enqueueAction(action);
    },
    createCasDetails: function(component,event,helper,responsedata){
        console.log("responsedata---"+responsedata.messageDisplay);
        if(responsedata.messageDisplay == false){
            var Acc = component.get("v.Preaccount");
            var con = component.get("v.Precontact");
            console.log("Acccase---"+Acc);
            console.log("concase---"+con);
            var createCaseEvent = $A.get("e.force:createRecord");
            createCaseEvent.setParams({
                "entityApiName": "Case",
                "defaultFieldValues": {
                    'Case_Type__c'     : 'Do Not Call List Request',
                    'Account'        : Acc,
                    'Contact'        : con,
                    'Opportunity__c': responsedata.caseOppId
                }
            });
            createCaseEvent.fire();
        }
    },
})

Apex Class:

public with sharing class RemovefromDNCcontroller {

    @AuraEnabled public string caseaccId{get;set;}
    @AuraEnabled public string caseContactId{get;set;}
    @AuraEnabled public string caseOppId{get;set;}
    @AuraEnabled public string message{get;set;}
    @AuraEnabled public boolean messageDisplay{get;set;}
    
    @AuraEnabled 
    public static RemovefromDNCcontroller Oppdetails (Id OppId){
        RemovefromDNCcontroller rfd = new RemovefromDNCcontroller();
         rfd.messageDisplay = false;
        Opportunity o = [select id, Name,ownerId, AccountId, Account.PersonContactId, Do_Not_Call__c from Opportunity where Id =: OppId];
        system.debug('o----'+o);
        if(o.Do_Not_Call__c == true){
            system.debug('rfd 24'+ rfd.message);
            rfd.caseOppId = o.Id;
            rfd.caseContactId = o.Account.PersonContactId;
            rfd.caseaccId = o.AccountId;
            system.debug('caseContactId ==='+rfd.caseContactId);
            system.debug('caseaccId ==='+rfd.caseaccId);
            return rfd;
        }else{
            rfd.message = system.Label.Remove_from_DNC_Alert;
            rfd.messageDisplay = true;
            system.debug('rfd 32'+ rfd.message);
            return rfd;
            
        }
        
    }
    @AuraEnabled 
    public static Id accountPreCheck (Id AccountcheckId){
        system.debug('AccountcheckId--'+AccountcheckId);
        return AccountcheckId;
    }
    @AuraEnabled 
    public static Id contactPreCheck (Id ContactcheckId){
        system.debug('ContactcheckId--'+ContactcheckId);
        return ContactcheckId;
    }
}

In Helper, method createCasDetails, values in 
console.log("Acccase---"+Acc);
console.log("concase---"+con);
are not coming due to which values in Account and Contact are not pre-populating.
Please help.