You're changing the subject to the author and not the idea. If that were something worth doing, I could simply state that Milton Friedman disagrees with Hayek and states the opposite and that that would be proof of my point. But that wouldn't make sense too. What one person says, even if that person is someone like Hayek or Friedman is no proof of something.Let me just say that "Constitution of Liberty" is not one of Hayek early works. It was written nearly 20 years after "The Road to Serfdom". Some of the points made are also reiterated in "Law, Legislation and Liberty", which was written even later.
Other than that, it doesn't appear that we're likely to agree on this, so I'll let a better man like Hayek speak on my behalf. Best of luck to you!
These men were logical men and logic dictates that we must look at facts. The fact is that government intervention or nationalization of services that can be regulated by free markets results in high prices and poor quality, compared to a free market system.
We indeed are probably not going to agree on the subject, but the issue is even more fundamental: The thing is that I have no objection with you living your life the way you stated. You want to live your life in a system which has government dealing with healh services? Fine by me. I wish you the best. The problem starts when you believe that you can force me to be a part of that.
And that is the fundamental problem with this. While I have no objection to your views and choices, provided you don't attempt to force me to be a part of it, the only way to apply what you are stating is by forcing me to be a part of it.

Best luck to you too!