It would have to be signed by both sides to count. I've done it before, but you can't just try to sneak things through if they don't sign.
From my understanding, the medical billing system is so fucked precisely because hospitals bought laws making it so they can arbitrarily bill patients without forming a proper contract (otherwise there would be no way to charge someone who was admitted while unconscious). In the absence of such laws they'd have to fall back to an unjust enrichment claim, under which it would be pretty hard to justify charging someone $25 for an aspirin.
If it ended up in court and they produced that form, would my amendments mean anything? Is it a contract when only one part signs it?