Not questioning if a human being deserves healthcare. Putting words in my mouth. It's a matter of who ends up paying for it. Ok, so according to you, the US taxpayer should be responsible for covering health care for basically everyone and anyone in the world who wants to come here, illegally or not? How in the world could that be financially feasible?
However, demetriosj, there is reason for the term 'universal healthcare'. Residents, immigrant residents, and refugees/asylum seekers (presumably) are entitled to receive care in the public health system. The concept of public health care is based on the view that care is a 'right' and the view that equal access to healthcare makes society better for all. And taxpayers contribute to the system believing that to be the case.
It is financially feasible for pretty much all countries in the western developed world - though many of those systems are in crises these days, especially given that populations are aging.
But I don't think you'll find anyone in those countries who prefers a private system, unless they are very wealthy.