Well, in the US as a whole, businesses can generally reject employment for any reason, no matter how petty. There's a small, finite list of reasons that are illegal -- race, sex, and the like -- but these have to be specifically enshrined in the law.
Local governments can add their own laws on top of this, of course. Discrimination against sexual orientation has become illegal in many places within the US.
The letter of the law is one thing - the recommendations of HR lawyers to employers wanting to stay out of trouble are far more conservative. The advice I've heard is basically "don't do anything to create or discriminate against a 'protected class'". So even if it's not strictly against the law to decline to hire or promote people who wear blue shoes on Tuesdays, it's probably a bad idea to make such a policy without a strong, well-documented, objective basis for it.
Local governments can add their own laws on top of this, of course. Discrimination against sexual orientation has become illegal in many places within the US.