You don't seriously think the judicial system will allow a systemically critical piece of society go bankrupt do you? If they won't allow banks to fail, you can be sure they won't allow hospitals to fail, at least not at the hand of a judge.
As for Herbalife, Icahn is already in the money on his call options. He can and, as far as anyone knows, may already have liquidated all or part of those calls at a profit. He is currently winning. Loeb has also been selling part of his position, according to various news outlets. That brings us to Ackman. He is only losing on paper at this point. If Icahn and Loeb sell out of their positions, that will leave mostly passive investors to support the stock. It would not surprise me in the least if Icahn and Loeb exit their positions entirely by end of calendar 2Q or, more likely, largely exit them but leave a token amount long, and if the stock declines significantly by calendar year end leaving Ackman with a substantial gain as well. None of them will lose money in the end.
That's where Third Point comes in. Of course, there are a lot of other players, but they increased their position substantially to lift the stock. Third Point will find a way out of this position but not until it extracts its pound of flesh.