-Porous borders and unwanted immigrants seems to be an eternal problem. -Your point pretty much rules out universal social insurance programs. -In 1938, you could have rejected racist Social Security on those grounds.
No-- because social security was replacing a reality of almost no significant pension systems.
The current situation is a patchwork of health care plans that imperfectly offer health care to a wide range of the population, even to the uninsured through walk-ins to clinics and emergency rooms. The law requires basic medical care even for the completely uninsured if they show up at the emergency room, even if they do get shuttled as quickly as possible to a dumping ground public hospital afterwards.
But I think there is a significant possibility that any single-payer system would exchange more universal coverage for citizens with the elimination of access for anyone without proper documentation. So for at least 10 million people, single payer health care would likely make their health care situation far worse.
This is one reason I actually lean towards incremental improvements of the existing system-- tighten employer responsibility, expand Medicaid, expand CHiPS, create low-cost funds for smaller employers, and so on. That fragmentary approach is likely to guarantee better coverage for more people than a "big bang" reform that might leave a whole range of people even more screwed.
Nathan Newman