Most/many of the homeless that come in are eligible for Medicaid or other government benefits (veteran healthcare for instance). However figuring that out can be very difficult. Many of them are incapable or unwilling to give up their names or stay long enough for social workers to get their paperwork in order. Or they're afraid of being located (for example they have warrants out for their arrest). Then you have regular uninsured people who are simply incapable of paying that come in and give fake information.

Fun fact, if you're unable to pay the full amount most hospitals will do almost ANYTHING to stop from having to send your bill to collection. Even if you say 'I'll pay $50/month forever' thats way better than they'll get from the debt agency that buys your debt.