"My boyfriend broke up with me because of X, Y or Z reason. It has been tough, but I've been following your advice, and I haven't contacted him at all since it happened. He texted me a couple times asking how I am, but I didn't reply. The thing is, his birthday is next week. It is OK to send him a message to say 'happy birthday'? I don't want him to think I am rude."The answer is definitively, absolutely, always, without a question "no."
I don't care that you've decided to "remain friends" or how close you still are. I don't care if he got you a huge present last year and you "feel the need to return the gesture." I don't care if you say "but we just broke up a couple days ago," or that "I always do that kind of thing for people I care about, and I still care about him." And I definitely don't care if you "just think it would be kind of mean not to."
Don't do it.
All of the above are rationalizations - mere excuses for remaining in contact with him, for trying to re-initiate something or seeing if he'll take the opportunity to re-initiate something given the chance. He isn't your boyfriend anymore, and as I explain in my book, he really isn't a friend or acquaintance anymore either. He is an ex, you're single, and as long as you are keeping strings attached to him you aren't attaching new strings to someone else.
1. Book Release: Beyond the Breakup
2. No, You Can't Be "Just Friends"
3. Stop Having Sex to Prove He Likes You