Should I wish my ex a happy birthday, or not? Good question. I think the answer has to be based on the answer to a second question which you need to ask yourself: "Does my ex hate me?"
You probably have a pretty good idea of how your ex feels about you. If, when you're honest with yourself, you have to admit your ex hates you, then the answer is no, you should not wish your ex a happy birthday. In fact, you shouldn't communicate with your ex at all, until you have reason to think those hateful feelings have changed. Saying happy birthday to an ex who hates you and doesn't want to hear from you is not a way to accomplish getting back together, if that's what you're after.
But otherwise, yes -- go ahead and wish your ex happy birthday. Just, don't do it in a manner that is too aggressive. For example, don't show up out of the blue at your ex's home, just to say happy birthday. Don't use an ex girlfriend's birthday or ex boyfriend's birthday as an opportunity to stalk.
If you are thinking about your ex's birthday in advance, it would be really nice to send a card. Sending a card communicates some positive things about you. It says, "I've got my act together, and I planned ahead." And it says, "I'm not desperate to talk to you." It makes you look like you're in control.
If today is your ex's birthday (or there isn't time to send a paper card), and you are likely to see your ex in person when you're just following your normal schedule, then say happy birthday when you get the chance. Otherwise you can send a text, or email an e-card, or call.
Calling is tricky, though, because you only want to call once. If your ex answers, good, but if not, you need to leave a message. Even if you know your ex doesn't check messages, leave a message anyway because this is the only call you get to make. If you hang up without leaving a message, you can't call back because you don't want your ex to think you called multiple times.