Such an agreement would generally be illegal, IIRC. I believe a bunch of tech companies got successfully sued for it. Basically making a cartel on the purchase side is just as illegal as a cartel on the selling side.
Non-solicitation and non-compete agreements are legally limited and restricted, but not completely illegal.
There was a case about this kind of thing just last year, where the precedent set was that a non-compete agreement
as part of a larger business collaboration deal can be legally valid where a standalone non-compete agreement is not. Essentially, if you're Company A and I'm Company B and we agree to a mutual non-solicitation agreement
and just that, that is not valid and we're both guilty of anti-competitive practices. But if we agree to a mutual non-solicitation agreement
as part of a business deal between us, where the idea is, "Because of the work we're agreeing to do together, you'll have direct contact with my employees and vice versa, so we both agree that as part of the collaboration deal, we're not going to take advantage of that to try to poach each other's stars."
I'm not saying there definitely is a non-solicitation agreement between the two companies. I'm saying that the way they each phrased their public statements might imply one, and that's one of the very few ways ThedaCare could possibly have any case at all.
Edit: And yes, non-solicits can exist in at-will employment conditions. For example, it's pretty much universal for temp companies to put a non-solicit clause on the contracts they make with property management companies, because if they
don't have such a clause, it's very common for a company that acquires an excellent temp employee to poach them as a direct hire.