This is simply untrue. Government run health care reduces costs (by eliminating the rent-seeking middlemen in the process and by controlling costs through direct fiat if necessary) and improves outcomes. Government run health care also incentivizes the government to tax behaviors and products that have negative long-term health consequences so that it can control the eventual costs that it will end up paying.
US health care is not the best in the world and has not been for a very long time. Some people may travel to the US to see a specialist, but 10x as many Americans travel to other countries to get care that they are unable to get in the US (usually due to cost.) The US usually ends up dead last in health care among the more industrialized nations and more recently was ranked 37th in the world by the WHO. Please find me a single ranking that puts US health care at the top of the list, I could use the laugh.