If you present a contract to the hospital and explain that these are your terms, and a reasonably understood officer of the business signs it, then sure: they're on the hook. Contracts are contracts, and the law gives great weight to consent, even in circumstances like this where it's not clear a real negotiation happened.

The idea above was about "swapping" some random document in for the standard form and presenting it to them as if it was their document. That's not good faith negotiation, that's just fraud.