You should look into it. It very much depends on the country.

But typically you just start paying taxes or fees or whatnot into whatever the national system is in place. Most countries will require you to have proof of some type of outside insurance before immigrating.

There are overseas specific health insurance plans you can get on -- typically not very expensive because healthcare in the rest of the world isn't insane. [1]

Most countries also have a cash-only payment schedule. Unlike the U.S., fees for health services are typically known up front.

In any case, even paying cash, you're likely to find healthcare outside of the U.S. (in tier-1 countries like the EU, Japan, Korea, etc.) to be high quality and very low cost -- often cheaper than healthcare in the U.S. with insurance.

1 - https://www.aetnainternational.com/