No, Texas is an employment at will state.
They can fire you for whatever reason they like.
Not if it’s unjust. And I would pursue the case to the highest court if I had to. It becomes a matter of money, and how much the company would be willing to spend. Since I have lots of it, it would only be a matter of time before they would cut their losses and settle, because winning will get them nothing at all.
But since I own my own company, that will never happen. :o)
Unless an employee sues my company. Then of course the shoe will be on the other foot. I’d have to decide what the cheaper thing to do is, pay them off, or take the case to conclusion.