Learn English – Birth date or birth day

An appointment is a formal arrangement to meet or visit somebody. To me, that means you and somebody else took a decision about when to meet; for example, that would be the case for a dental appointment.
In your case, I would use next meetings.

As side note, in British English date also means "an arrangement to meet somebody at a particular time"; in all the English dialects, date also means "romantic meeting."

In everyday language "birthday" is associated more with the anniversary of the date of birth. When asked about it, a person may (or would likely) provide just the month and day, but not the year.

"Date of birth", however, is much more difficult to misinterpret.

Whether to capitalize "birth" in your User Interface is probably up to your UX designers.