When they put the hat on his head he comes to life and exclaims, “Happy birthday!” I always thought he was saying this to Jesus Christ. My son thought he was talking about himself but who says “Happy birthday!” to themselves? At least without adding a “to me!” in there.
Why does Frosty say “Happy birthday!” when the hat is placed on his head?
Maybe because he just came to life so it’s his birthday, but more likely because he’s naive and mixed up, and not very bright. Not his fault, as he has snow for brains.
This has always been my take on it. Frosty is a friendly and gregarious dunderhead, and “happy birthday” is a friendly and gregarious sort of thing to say, that can be spouted at any time if you don’t think too much about the implications.
I seriously doubt it was intended as any kind of slantwise religious reference or commentary on the beginnings of individual existence. It’s much more likely akin to the reason Scooby Doo can talk. You don’t want to be Mysteries Incorporated. 
And he has snowballs.
Because it’s his birthday. He’s a simple and literal soul. There really aren’t many Christmas references in the show IIRC.
Except for the whole “going to the North Pole so Frosty won’t melt and him melting anyway to save the little girl from freezing by carrying her into Santa’s greenhouse” thing.
So does Robert Downey Jr., but it doesn’t stop him making millions.
I’ve always wondered why he needs to wear a hat and scarf. To keep him cold in the sun?
Well, we can’t have him running around naked, can we?
Perhaps I should have said no overt references to Jesus. That would have been much more accurate.
It probably should have been made clear in the OP and title that this was about the TV special, not the song itself. I consider the special to have been not that good. The song, however, is quite fun. 
I always took it not as a blessing (“Happy Birthday [to me]!” or “Happy Birthday [to Jesus]!”) but as a joyful declarative: “[It’s a] Happy Birthday!” It’s his birthday, he’s just been born to life, and he’s happy about it. And why shouldn’t he be.
Wishing himself a happy birthday is a possibility, but I don’t think it’s because he’s not bright. It’s even less likely to be a nod to Jesus.
I seem to recall that the original song ended “I’ll be back again someday,” but that the TV special changed it to “I’ll be back on Christmas Day.” Is my memory correct? That makes no sense. It doesn’t necessarily snow on Christmas!
But as I recall, Frosty was made of “Christmas snow,” which was part of the magic. It also happened to be the first snowfall of the year, making it doubly magical. At least that’s what I remember Jimmy Durante saying. Thus there is some connection between Christmas and Frosty coming to life, so it makes sense that Christmas would be the day for Frosty to return.
For the same reason men and women wear earrings. It looks cool 
This.
I just always thought he said it because he wasn’t all that smart and was confused that he had suddenly come to life.
But I haven’t seen the animated special in decades, and didn’t like it very much when I was little. They always put Frosty on after one of the really cool stop-motion animation ones like Rudolph or Santa Clause is Coming to Town, so you were sorta forced to sit through Frosty in that pity time slot.
At least Frosty didn’t try to kill everybody.
NSFW - “Jesus vs. Frosty” - South Park precursor by Trey Parker and Matt Stone.
This was Frosty’s safe word.
iirc, and it has been many a moon since I’ve seen the Frosty the Snowman TV Special, he says “happy birthday” because the magic top hat is the hat of a magician performing tricks at a child’s birthday party. The magician is in the middle of a trick using the hat when the wind somehow liberates said hat from the magician mid-trick and yadda yadda and there you have it. The “Happy Birthday” is the magical stutter of “majika interruptus”
in the matrix its a deja-vous from the system resetting to correct an error
I never really watched the whole thing, because even to my 12-year-old self it was dopey, but I instantly recognized Paul Coker, Jr.'s animation style from MAD magazine.