They'd have to prove you disclosed it. As has been discussed here recently, there are plenty of ways to do that among colleagues while maintaining plausible denial.
If it's at-will, they can fire you for whatever reason (or no reason). I don't think they have to prove anything, except that you signed an at-will employment agreement.
Sure, but if the word on the street was that Company X fired somebody for something as petty as disclosing salary, which again is not enforceable in most States in the US, then I'd love to see them be able to hire effectively in this market.
