In countries where they have free health care, I would be willing to bet that people are treated better than there are through our Veterans Administration. Also, many people with health insurance die because their insurance provider digs up some excuse to deny their coverage. Next, taking something for granted is a small thing. Becoming homeless because you can't afford to pay for your medical treatment is a BIG thing. I would say that one couple who becomes homeless because they can't afford to pay for their medical treatment is equal to approximately 316,443 people who take free medical coverage for granted. That is "if" they take it for granted.
Well you would lose the bet because they aren't. That's not to say the VA has the greatest health care man has ever seen. The VA is actually the system you want for everyone, which is a government-funded system that is sub par to it's private sector counterpart.
No one becomes homeless because they can't afford health care in America. Every state in America has indigent care laws which prohibit hospitals from denying treatment due to inability to pay. Every county in America has a government-funded health clinic with pricing established according to income. Every disabled person, poor person with children or retired person has Medicare and Medicaid who pays for their medical care.
Meanwhile, in Canada... one of those "other countries" often cited in your pipe dream of free government health care... people wait for a routine MRI an average of 6-9 months as opposed to a couple of weeks in the US (prior to Obamacare). Open heart surgery-- 3-5 years as opposed to weeks (at best) in the US. If you need a kidney or organ transplant in Canada... better get your will in order.
What you will ultimately get with "single payer" or government-paid health care is rationing of health care. There is no other way to do it. So you are going to have people who are told they can't have their hip replacement or heart stints, it's not practical because of their age. And you think insurance companies are a hassle to deal with now... wait until it's a federal government bureaucrat on the other end of the line. When you are being mistreated by a private sector insurance company, you have the recourse of your state insurance regulator and attorney general... when it's the government, you're shit out of luck. You take what you get.