I have added support for birthdays with commit 6ee010905e172164b865c0e9c0576d92d1ae7906, but I can’t test it because my Android 4.1 and lower devices don’t show the birthday anyways. Closing this as duplicate of issue #5, please post there.
Generally, it’s possible to sync all properties listed here: https://developer.android.com/reference/android/provider/CalendarContract.html
But it depends on the Android version, and often its not possible to translate the Android values 1:1 to vCard values.