They used to use the account from the Bible where John the Baptist was beheaded and his head was presented on a serving platter at the birthday celebration of Pharaoh's Daughter.
I never saw how that account meant that we shouldn't celebrate birthdays. What I took away from it was that we should never behead anyone at a birthday celebration and we probably shouldn't use serving platters since the Pagans apparently used them.
For that reason, we always used to acknowledge our boys birthdays by recanting the story of the day they were born and getting them a little something special even if it was only a special dessert at dinner, just for them.
Now that they're grown up and the Watchtower is a thing of the past, I'm glad that we went with our gut on that one (a few others as well) and we weren't robbed of yet another harmless life experience.