I don't really see how this can be a question, other than perhaps the OP is wondering who has the moral high-ground in their post-relationship contact. A birthday is a day for feeling good about yourself, and for other people to support you in this. Birthdays are important to people. For one day, you get to feel like it's OK to be a little bit selfish or pampered, or to just do something for yourself and let your obligations rest for one day.
If break up with someone on that day when: a) you have another 364 days to choose from; b) it will make no difference you whatsoever if you broke up the next day; c) the importance of the birthday to a person that I just described, you are acting in a cruel way. Perhaps this is tit-for-tat for them having treated you cruelly in the past, but you can't get away from the nature of your own actions.