I don't think it would make healthcare costs unsustainable if we actually secured our borders and focused on making the system we set up efficient. Pretty much every other first world country has figured out a way to pull it off. If Europe can figure out how to make universal healthcare affordable under capitalism so can we. We're a smart and innovative people when we come together and work towards a goal.
But it isn't affordable in Europe. Or Canada. Or anywhere else.
A very good friend of mine in Wales had a relative with a curable cancer. His diminishing health and high health care costs soon left him broke, and along with it, abandoned to die by the national health system. The cost for treatment was determined to be too high...
He figured out, pretty much with his last gasps, how to get to the USA, landing in Florida, and immediately became indigent. American welfare provided for him, and in 8 months, he was back to health, off of welfare, and applying for American citizenship.
Another friend in Canada had terrible health problems and was faced with huge wait times - He required a relatively mundane intestinal surgery, but would have died, had he not taken action that the Canadian system would not allow... His family ponied up the money and got him down to the states, where the American system provided the needed surgery.
Be careful what you wish for - All that glitters is not gold.