I've heard horror stories about hospitals not accepting insurance. Wouldn't want to be in a situation of being ill and having to pay more than I can earn in a lifetime.
They will accept it. The cover has to be specifically for US hospitals otherwise the insurer won't pay out and they know that and won't accept it. You have to avoid insurers who only cover certain providers as well.
You have to read your insurance contract and info sheet properly rather than go for the lowest price.
As for the medical costs - if you want to be on the safe side, you can (actually you should) get travel health insurance.