I often hear from employees who are bound by non-compete agreements that "Texas is a right to work state," and that, therefore, their agreements are unenforceable.

Not true.  The "right to work" concept has NOTHING to do with non-compete agreements.

Rather, the "right to work" concept has to do with the right of employees to decide for themselves whether they will financially support a union (assuming there is one).  It has nothing to do with whether non-compete agreements are enforceable.  Nothing whatsoever.

So, if you are bound by a non-compete agreement and someone says, "That’s unenforceable–Texas is a right to work state," ignore them.