Yes, although they are at least actually in the country! I believe actually there have been a few attempts to claim refugee status due to lack of health care, but it failed to meet the criteria. Normally citizen, refugees, and landed immigrants have access here. Although I have never known people to be turned away - one family who had no access went to the clinic I worked at, and their doctor never got any payment. They used to bring him baked goods though.
The case in the US with illegal immigrants seems complicated with no adequate solutions. On the one hand, they seem essential to keep the American economy running. If every illegal working in America was sent home, there would be a lot of empty jobs, the ones most Americans don’t want.
As far as I know, no one has a good effective solution yet for the problem of illegal immigration and its associated economic issues.
But once they are there, working for American citizens, for wages American’s wouldn’t accept, what do you do about their health care? (or workers rights, etc…) It seems that those who hire illegals should be responsible for their health care, but that just doesn’t work, for obvious reasons. Leaving people with no way to access care also seems wrong, especially when society to some extent turns a blind eye to their existence.
It’s a problem, but I have no clue what would be a better approach.