There is no such thing as an idle threat by a debt collector. Collectors are not allowed to make statements about actions they will take unless they both intend to take those actions, and have the legal ability to do so.
In your case, however, it sounds like you are dealing directly with the creditor (the hospital) and not a collection agency. if so, the Fair Debt Collection Practices Act does not apply, but other state laws may.
You can be sued if you do not pay your medical bills. If the hospital gets a judgment against you, they may be able to get a lien on property you own. It's a matter of state law, and the best person to advise you there is a bankruptcy attorney who will be familiar with this type of situation.
Given what you're telling us, we would encourage you to at least set up a free consultation with a bankruptcy attorney to find out whether the hospital can go after your home equity, and if so, what you can do to protect it.