>>19956188omitting it would be 100% wrong
"a" is grammatically correct but not the best choice in that specific context ("he has a birthday on 4/20"), but you do need some sort of determiner there.
A birthday is a day when...
My birthday is next week
Whose birthday is it?
Two birthdays at the office next week
It's not a proper noun. It can't be treated the same way as actual holidays
