Earlier quoted context omitted.
Care for them under the universal system, but bill the cost of the universal system directly.
My mom got injured while hiking in a country with universal care. She hobbled to the next town, found a clinic, and got treated. Then she wanted to pay. First, the person at the clinic said: "What do you mean, you don't pay for health care." My mom explained that she was from another country. It turned out they didn't even have the means to take her money. Treating a few foreigners for free might actually be cheaper…
The US has no such deals because it does not have a UHC system to deal with. However, most UHC systems that do not charge aren't going to pay out for setting up a billing infrastructure for the occasional American. The ones with co-pays etc already have a billing structure in place, and may charge you. Or not, it would be pretty mortifying to have to charge someone for health care.