As someone who lives in a country with a public funded health system, I can definetely see Why many americans are against the implementation of anything similar in the USA.
The thing is, there is no "right" that government intends to guarantee people, that does not mean taxation, and when it comes to public health system, it's usually high taxation. People are heavily taxed for a service, that often doesn't work. If they need long term treatments or more specialized exams, they better be prepared to pay for them in the private sectors, meaning that, despite having paid many times for these things in the form of taxes, people will have to seek a private services if they don't want to wait a year or more before it's your time in the line. Sometimes, depending of the type of help people need, they just can't wait.
I mentioned the "line" that contributors need to take before using the services they have paid for (many times, often) and that's another aspect of the problem with public funded health. The "social" character of the public system, guarantees that, even people who don't pay taxes and live of social programs, have the same right to use the system as people who actually contribute, meaning that the ones who do pay the taxes, also pay for the exams, medicine and treatments of others who never contribute, and usually these people who don't contribute have priority in the use of the benefits of the system (Called SUS here, which means: Unified Health System). Another thing, the costs to maintain a public health system can only up, when we consider the trend of western civilization of having an aging population. Govern often has to cover the losses of the system, increasing taxes or taking money from somewhere else (more taxes too).
All in all, I am not against some sort of public health system. I just see the need to reduce the scope of services the system can offer. Maybe just concentrating on emergency treatments, distribution of medicine and imuzation through vaccines. Contributors should keep the money they would pay to the govern for themselves and seek treatment wherever is better for them.