Keeping my birthday private
Your birthday will be private. Read the privacy policy if you're concerned about it.

If your birthday appears in your calendar, it's probably because your birthday has been added to your private card in Address Book. There's an option to show birthdays in Calendar. However the birthdays are only showed to you, not your contacts.

Beware of entering an erroneous birth date. Apple can ask you your birthday to verify you're the actual owner of the Apple ID for security purpose. So if you enter a false birthday, be sure not to forget it.