You can file this next to my previous immigration compromise, which liberals and conservatives can both hate in equal measure:
Whatever country in the world the U.S. should take as its model for national health care, the U.S. should also take as a model for immigration policy.
You want Switzerland's very solid universal health care? Then you take Switzerland's very restrictive immigration laws. You want China's really cheap universal health care coverage? Then you have to take China's incredibly stringent immigration laws.
(In order to get a visa to work in China as a teacher, I was required by the Chinese government to have a full physical exam, complete with an HIV test, TB screen, and an EKG. No one with a heart condition is allowed to immigrate to China to work. I had to demonstrate to the government of China that someone in the government of China had officially invited me. Now THAT is a tough immigration policy. China won't even let overweight people adopt babies from China. Definitely hard core.)
Let liberals import another country's health care system, and let conservatives import the immigration policies. Unless we're saying that not everything Europeans do should be emulated.....