Hi there,

No idea exactly why that's happening, because I don't see any references to date literals not being supported with custom date fields. I did confirm the results of your test, and then looked for a workaround. I tried doing various >, >=, and other operators on both literals and constructed dates, and it stubbornly refused to filter on the custom date field as you would expect. I did find one thing that worked, though you'll have to evaluate whether it will work for you. If you make a Birthdate__c field of type datetime (instead of date), then using a date FUNCTION instead of a literal.  

Date d = Date.today();
for (Contact con: [
        SELECT Id, Name
        FROM Contact
        WHERE CALENDAR_MONTH(Birthdate__c) = :d.month()
        ORDER BY Name
        LIMIT 10
        ]) {
    System.debug(con);
}

The larger issue you're going to hit, though, is that even with deterministic encryption, you can't filter by date in a query. Once your new field has been encrypted, it will bark at you that it can't be used in a where clause. Strings can be used in query where clauses when encrypted with deterministic encryption, but that won't help you with dates/datetimes. You might be able to use something to store dates as a string in a way that would let you use them programmatically, but your end users using list views will still have problems with the solution. Another thing to consider would be a formula field that abstracts this complexity for your users, as they will be forced to update their list views to point to a new field either way. Performance might take a hit, because formula fields are evaluated at runtime, but you'll have to decide if the tradeoff is worth it based on the specifics of your org/use cases.

Wish I had a better answer for you, but it may boil down to educating your end users on the tradeoffs of encryption and the limitations of deterministic encryption: 
https://developer.salesforce.com/docs/atlas.en-us.securityImplGuide.meta/securityImplGuide/security_pe_deterministic_considerations.htm
https://developer.salesforce.com/docs/atlas.en-us.securityImplGuide.meta/securityImplGuide/security_pe_considerations_general.htm