You'd be surprised, English.bloke. I know people with little coverage or no coverage who are convinced by the anti-nationalised medicine/insurance propaganda out there. I am not sure if any of them would go so far as to protest, but it wouldn't shock me.
ETA: My mom who just got Medicare (she was uninsured before that). I am not sure if she is paying for extra coverage with that, but even with extra coverage, most people don't have great insurance with Medicare. She is totally convinced that we wait an eternity for treatment here. Every time I need to see the doctor for something, she asks me how long it took to get the appointment, what was covered, etc... Every time I explain that we can usually see a GP the next day and usually don't wait more than a week or two for specialists (longest non-dental wait was 6 weeks for a physical therapist). She seems to totally put that out of her mind.
It's human nature to resist changing perceptions.