iPhone Birthdays vs OSX birthdays

On the mac, turn off Birthday calendars in the Calendar app preferences.

On the iOS device, unselect the check next to the programmatic calendar to suppress those entries. Short of jailbreaking the device and patching the Calendar app, you can't suppress the package calendar - just show or hide it.

Alternatively, you could purge all birthdays and anniversaries from all Contacts, but that seems too brute force to me when you can suppress each calendar with settings. Most that I have asked choose either the OS X calendar and suppress the iOS one or skip the OS X one and rely only on iOS for birthdays and anniversary entries.