I was told it's also illegal in the US for a hospital to bill you as uninsured if you have insurance.

You won't get in trouble but the hospital will, but if they ask if you have insurance and you say no when you do, that could change the situation.

Again, I'm not a lawyer but this was they told me once a few years ago because I got charged a ridiculous amount for something and wanted to see if it would be cheaper if I just paid without insurance considering my deductible was many thousands of dollars.