What if you were born on a different day than you really were?[/b]
(emphasis added)

Then you would still be you. Your use of 'you' in the question already assumes that

But if you phrased the question differently, like "What if your mother gave birth to someone in a different time/place/way than you, instead of the way you were born? Would that person be you?". Then, hmm, probably it would not. That person would almost certainly grow up to have different life experiences, and so result with a different personality, opinions, ideas, skills, etc, even if they had the exact same genetic make-up. If their genes were different, then they would naturally form physically differently, so they'd be somebody else for sure.