Ideally you would provide a date field (with date picker dialog) plus a switch where the user can select, which part of the date can be displayed (full date/just day&month/just month&year/not at all).

This way the birthday can still be imported from a central directory but the user is empowered to decide how it can be used.

This would be very beneficial when celebrating birthdays, as an example. We would like to record the MM-d but not the year.