Rather than take the side of an unpopular position on the medical care issue and oppose universal medical care, I would suggest that immigration restrictionists specifically oppose the subsidization of medical care for non-citizens on the part of citizens. There is no reason why the employers of illegal aliens(or investors in those employers) cannot be held 100% accountable for the cost of medical care of non-citizens. If need be, we can specifically increase the minimum wage of non-citizens to pay these extra fees. Now, many employers of illegal aliens don't properly file taxes-but when they are discovered, property of the employers of illegal aliens could be seized-as could the property of the landlords that rent to illegal aliens and those that invest in businesses that employ illegal aliens. After all, those that get caught need to pay for health care expenses of those that aren't caught-so I'd expect these fines would have to be pretty high for the system to be self-supporting.
Now, it would be rather difficult for businesses hiring folks at $5.50 per hour to pay for $1500/month in medical and dental benefits for a family of 5. We'd see a lot fewer folks employed in those positions if such costs were imposed. Illegal immigration is inherently a highly-subsidized activity-without those subsidies, much of it would disappear.