Can a college student claim some of their scholarship money is being used for room & board if they live in an off-campus apartment vs. on-campus housing?

No, he cannot, unfortunately. Room and board is not a qualified education expense.  Scholarship used to cover these expenses is considered as taxable. 

I am aware that scholarship money used for a non qualified education expense is taxable and needs to be shown as unearned income on the student's tax return.  I just need to know if off campus housing is still considered room and board for a college student or if they need to live on campus to count it as college room and board.  Please advise.

Yes, it can apply to off campus room and board as well unless the scholarship specifies it's for on campus.