I've had this concern for a while myself.
I once quit a bay area tech startup job in my 20s due to burn out (5 years no vacation, easily 50+ hour weeks) and in those few months of taking time for myself I got hit by a bout of what I assume was appendicitis. I delayed going to the hospital because I didn't have insurance since I had just left my job. The pain passed that day, but then a few weeks later the same excruciating pain came back -- just really indescribable pain -- to the point where I semi-consciously was asking my partner to make sure I didn't die. Ended up at the hospital, they confirmed appendicitis, did the surgery in less than an hour, and I got hit with a $25k bill. I remember an administrator coming into the room before the surgery and asking me for a credit card while I was completely out of it, because I assume someone told them I didn't have insurance.
After it all happened I was talking to a friend of mine who is an ER doctor and he told me, quite frankly, that if you ever find yourself in a situation like that to just leave your wallet at home and give a fake name. As a doctor they're going to treat you no matter what.
It's horrifying that we have to deal with this system in the United States. It does make me consider seeking citizenship elsewhere, which I consider more seriously the older I get.