I believe that as long as you care about someone in your life, you should show them that you care. In your case, I don't think that you should simply wish butyhday greetings. Put in some extra effort. You could make the person an arts and crafts gift but if that isn't possible maybe a hand written happy birthday letter is the next best thing. Think first of the reason that you broke up then rectify that before trying to reboot a relationship.
If someone dumped YOU then it's on THEM to get back to you and win you over. Trying to get someone to be with you who just left you is one of the most delusional and biggest mistakes a person can make. She isn't going to want you back till the day SHE decides and realizes she made a mistake and you making it obvious you're still hung up on her won't make her like you more, it just makes you look like you can't move on.
Yes you wish them a happy birthday to show them that you remembered and that you were thinking of them even though you guys broke up with each other.