Well just adding few info why its not working!

In VF you cannot rerender a component which is not visible(rendered ="false") in page. Initially "appTypeDiv" is not visible because the condition doesnt match, SO to make this component visible we will have to rerender the parent component(which is visible). In our case its the "panelGrid"

So the code should be

<apex:actionRegion >
                        <apex:selectCheckBoxes value="{!selectedApps}" required="true">
                            <apex:selectOptions value="{!apps}"/>
                             <apex:actionSupport event="onchange" rerender="appTypePanel"/>
                        </apex:selectCheckBoxes>
                       </apex:actionRegion>

<apex:panelGrid columns="2" id="appTypePanel">   
                <!-- input field code for Cool App2-->
                <!-- input field code for Cool App3-->
                    <apex:outputPanel id="appTypeDiv" rendered="{!products = 'Cool App 1'}">
                     <apex:outputLabel for="appType"><span class="required">* </span> App Type:</apex:outputLabel>
                     <apex:inputField value="{!Lead.App_Type__c}" id="appType" required="true"/>
                    </apex:outputPanel>
                  </apex:panelGrid>

Ok ..my code didn't work but there is just too much to show here so here is a similar example that is same conceot from the sfdc docs but how come my <apex:outputText doesn't display after I select US checkbox?  i think if i can get this to work i can do the same type of logic to my other code

public class sampleCon {
    public String[] countries{get;set;}
    public PageReference test() {
        return null;
    }

    public List<SelectOption> getItems() {
        List<SelectOption> options = new List<SelectOption>();
        options.add(new SelectOption('US','US'));
        options.add(new SelectOption('CANADA','Canada'));
        options.add(new SelectOption('MEXICO','Mexico'));

        return options;
    }

}

<apex:page controller="sampleCon">
    <apex:form >
    <apex:actionRegion >
        <apex:selectCheckboxes value="{!countries}">
            <apex:selectOptions value="{!items}"/>
             <apex:actionSupport event="onchange"  reRender="myDiv"/>
        </apex:selectCheckboxes><br/>
    </apex:actionRegion>
        </apex:form>
        <apex:outputPanel id="myDiv">
            <apex:outputPanel rendered="{!countries= 'US'}">
               <apex:outputText value="The unformatted time right now is: {!NOW()}" />
            </apex:outputPanel>
        </apex:outputPanel>      
</apex:page>

I wil try this but the thing is, i only want to show or hide this field below if  one of the checkboxes is selected but the 2 other fields should always display. In that case, how should i do it since th panel grid is still there

                     <apex:inputField value="{!Lead.App_Type__c}" id="appType" required="true"/>