Surely more people neglect their health (as in, they may worry about it, but they don't do anything) because it costs money they don't have to go see a doctor?
I was a military wife when I was diagnosed with a serious, incurable condition that is very expensive. Most of my medical care was free. I had the occasional small copay for drugs not carried by the military hospital.
I began getting better by spending $300/month on supplements and making dietary changes.
I was a homemaker, so it "wasn't my money" and my husband wasn't an ass about it, but money was always a sore point and I always felt very aware of some of the psychosocial stuff where everyone not only thought I was kind of a loon for pursuing "alternative" remedies, but I was also "being financially irresponsible" to do so.
I got better. I feel justified. The world continues to call me a loon to my face.
I can't win for losing, but it's made me very aware that if you give free drugs to people, some people will take drugs instead of doing things like eating healthy and exercising.
I'm for universal coverage because we know health costs go up when people can't get certain basics and we know this actually puts people at risk of homelessness.
I worked in insurance for a few years. I worked for Aflac. They mostly do supplemental insurance and they made most of their money in Japan, a country with universal coverage.
So some countries provide universal coverage and allow for supplemental coverage. My best understanding is this works fairly well and fixes some of the problems I've heard Canadians complain of in their system, though I could be misinformed. I don't have first-hand experience with either.
One last argument that I think isn't entirely crazy talk:
People who can get free care may neglect their health because they figure "it's free." This can end up being a big burden on not just the healthcare system but society in general for various reasons.