I don't know the situation between you and your ex girlfriend, and I can't make a suggestion for you because of this. My advice would be to go with your gut feeling. If you never talk, an innocent 'Happy Birthday' probably won't make much difference. If you're still friends then I'm sure she'd have absolutely no problem with you greeting her. Try to imagine yourself in her situation and think whether she'd like it or not. Hope this helps!
We lived together for 5 years and ended it last august 2013 because she's afraid her family would find out about us (we're both girls). She has a boyfriend now but she still message me saying she's the happiest when she's with me, that they keep on fighting because of me, and that I'm the one she wants to grow old with. BUT she never tell me she wants me back. I'm trying to move on but she's making it hard for me to move on. I know I'm getting there but she's confusing me. I don't know what to do.
That's very unfair to you that she keeps contact and is confusing you. Your break up is unique in the fact that your relationship ended because of what other people think and feel and not what each of you think and feel toward each other. You should first decide what is best for you, and what you want. If you want or need to move on from her, then you need to cut all contact for at least one year. Which means no "happy birthdays" to each other. You both need space and time to heal if you truly want to go your own ways.
If you want to be with her AND you think this is the best thing for you, then you need to ask her where she is and what she wants. I want to stress the point that it needs to be the best thing for you because I'm sure you want to be with her, but that does not mean it will be a good thing, especially if she intends to keep your relationship a secret. You should figure out your position on the major problems you encounter in your relationship. Personally, I would not be okay with a significant other who is ashamed or embarrassed of me. Figure out what is important to you in a relationship, then ask her where she stands on those points. If she is not willing to work with you on what you consider to be important, I would suggest moving on and stopping all contact with her.
It depends on why you broke up. if it was an ok ending then I don't see why it would be bad to say happy birthday.My evil/funny side is if it was me and she hurt me I would write on her facebook saying "Omg I didn't know it was your birthday, thank god for facebook reminders"... happy birthday"'Up to you. Wont end your life if you do or don't. :Pgood luck