Most UHC countries have reciprocal deals where a citizen of one has all the same rights as a citizen of the other when travelling. Its just a lot cheaper than having to ship people around or delay treatment or all the paperwork.

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.