No one knows when Shuji's bday is except Shuji himself and the HR department at the Kirijo Group. He, obviously, doesn't talk about it much and doesn't celebrate it at all, at least not publicly. Maybe he buys a slice of his favorite cake and eats it with a nice cup of tea. This is good enough for him. And when it comes to when his actual bday is I like to do the Katekyo Hitman Reborn thing when I transcribe their names as numbers, which, in Shuji's case, gave me two options:
January 4th (「い」->「一」->「１月」 and 「しゅ」->「死」->「４日」)
September 1st (「いくつき」 stylized to read as 「一(日)九月」)
+ one extra: April 4th (the edgy double 「死」variant)