No shoes, No shirt, is a government health rule for food contamination.
as far as i am aware, it's nothing more than an "urban myth" that there is any state or federal weight behind 'no-shirt-no-shoes-no-service' rules.
there may be some local (city) ordinances but i don't know of any.
have a look - The Bare Facts No Shirt, No Shoes, No Service May Be The Rule But It Isn't The Law - tribunedigital-mcall
businesses may point to health concerns to defend their right to refuse service, but the rules originate purely from the business owners, not from the government.
the only thing that prevents any private business from refusing service to anyone in this country is the Civil Rights Act - and the amended version we have these days includes sexual deviancy as a protected class.
this bakery case isn't about refusing to serve a protected class altogether though; it's about a particular service. if they refused to even make a 'happy birthday' cake for such people, we'd have a much different set of circumstances, right?