I am not sure of your question, but can assure you that if the county owns and runs an organization facility that it is government entity. I am not sure about the collection aspect. If you have a bill I recommend that you contact a lawyer in person to see if they can assist you with the resolution of the matter.
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It either is our it isn't. It cannot pick and choose when it wants to be because it suits its purposes. So yes, it can offset your refund, but it can also be sued as a state actor, for example. If it weren't a governmental entity, it would have maintained this in court long ago.
A county is a unit of government, and I can see nothing in your facts that would show otherwise. If you have a bill, pay it, or if you cannot or dispute it, see a collections attorney. Often, bills from hospitals can be negotiated and reduced or can be paid over time. If you used services, you have a moral obligation to pay for them.
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