Can a hospital send me to collections if I pay my balance every month? How?

Yes, they can do this. Request document concerning their repayment policy. You can place the balance due on your credit card and then take as long as you wish.

This is not a country run by debtors. You are obligated to pay your account in full except to the extent that you carry your burden of proof to show that some other arrangement was subsequently created. You might be able to prove that objectively the debt was affected by an agreement to pay monthly indefinitely. If you want to pay merely the minimum payment and thus pay down little or no principal over time you'll be laughed out.

They have to sue. You can ask the judge to put you on a payment plan. Tell the hospital lawyer to work with you or you will file for bankruptcy.

A deal is a deal, and you promised to do what you promised to do. However, I'm willing to bet if you tell them you can't come up with the payoff, that they just might let you continue to pay. If you are able to pay a lump, you might want to ask them if they'd give you a discount to pay it off.

I doubt there is any legislation on this subject and of course there are plenty of laws around which did not arise from legislation. (It's called the Common Law.) You may wish to consult an experienced lawyer. It seems to me that he or she is likely to tell you that without some agreement in writing, or memorandum passed between the parties, you might defend against a collections lawsuit by pointing out that you had been making regular payments for quite some time without any complaint, or warning about a two-year limit, and that course of conduct between the parties should prevent the creditor from speeding up payments. (You might refer to the unstated agreement as an implied contract, and that by accepting your payments the lender and you reached an 'accord and satisfaction.' I am not certain this word work, but judges who care about doing justice (and some do) would not be pleased with the lender's conduct. Or you could borrow money from another source to pay off the balance of the bill, if you can. Or if you have lots of problem debt you might consider a bankruptcy. (But don't file one before consulting a well-informed lawyer.)

They can make any kind of demand they want. Tell then they're just lucky to get any money out of you at all, given that there are probably hundreds, maybe thousands, of collections they have that go unsatisfied. Continue paying as you have as your measure of good faith. If they give you a hard time, hire a lawyer.