If something is a “right”, it doesn’t imply the government should just take care of everything and it should be totally socialized. One might consider “clothing” a right, but I don’t think we’d want the government to control what we wear.
I agree that government should ensure universal healthcare, and maybe a single payer system would be the correct model (though other countries have had better results with partially market driven approaches). But my belief there is based on the fact that humans aren’t currently able to make good decisions about their own health care, and government panels probably would be able to make better decisions regarding public health.