I must respectfuly disagree with you both in the case of a business open to the public. You can't deny a whole class of peoples civil rights, whether it's a sign saying "we don't serve negroes", or a sign saying "we don't serve gun carriers".
While I think that only the vile and ignorant are racists,(I have a bit of personal experience with such.) I also think property which you have bought and paid for should be yours to use in any way which you see fit, with very few exceptions.
A free market has a way of sorting such things out on its own.