This question comes up fairly frequently so I thought I'd answer it now and get it over with. I am typing this on July 4th. The same applies for most (but not quite all) holidays.
There are four possible groups for the purpose of this question - (a) exempt employees who worked on the holiday; (b) exempt employees who did not work on the holiday; (c) non-exempt employees who did not work on the holiday;(d) non-exempt employees who did work on the holiday.
In all cases, if a legally binding and enforceable contract entitles you to something that it says below you are not entitled to, your contract holds.
(a) no matter what state you are in, you are entitled to be paid but you are not entitled to any kind of premium for working today.
(b) no matter what state you are in, you are entitled to be paid IF you worked at any time during the work week. Your employer may legally require that you use vacation time or other paid leave time for the holiday. If you did not work AT ALL during the entire work week, then you do not have to be paid for today.
(c) No matter what state you are in, you do not have to be paid. That doesn't mean you won't be paid; many employers offer paid holidays. But if your employer doesn't, the law is not going to force him. Your employer may legally require that you use vacation time for today.
(d) If you are in any state except Massachusetts or Rhode Island, you do have to be paid but you do not have to be paid any premium for working on the holiday. If you are in MA or RI, then you have to be paid, but your industry determines whether you need to be paid a premium or not. Some employees do; some employees do not. If you are a non-exempt employee in MA or RI and you worked on Independence Day and you want to know if you are due a premium or not, post below WITH YOUR INDUSTRY so that we can do further research.