You never mention what is his actual birthday and if he has a birth certificate what does it mention as the actual birthday.
If for any reason he catches the attention of the Customs/INS agents as he is re-entering the country, the inconsistent birthdays can become a real time consuming problem. In-consistency on anything alerts us anti-terrorist types to look very hard at the traveler for illegal entry, false representation, possible imposter etc.
Trying to get the naturalization certificate changed is possible if a) it is wrong, and b) if the date change does not trigger a change in status. Issues like how was he/she naturalized, via parents, or quota, or . . . can result in an “un-naturalization”.
My experience is that when people use in-consistent birthdays, they are trying to hide past deeds.