Nationalizing health insurance might improve quality; the quality of health care in the US is pretty appallingly bad, compared to, say, Argentina. I hear health care in nearly all the other OECD countries is also nationalized, lower in cost, and higher in quality than in the US, but I only have personal experience with health care in the US, Argentina, Canada, and the Federated States of Micronesia (where I can assure you that health care is terrible).

Admittedly the health care we've used so far here in Argentina hasn't been the public system, but I hear it's not that bad either.