The government health system in Aus is similar to the canadian and british one. People do have the choice to be in private health insurance or to be in the government scheme.
I prefer the government scheme. I was involved in a serious car accident a few years ago. I had private health cover, so in order to pay for my $165000 hospital bill, I had to sue the poor man that hit me (it was a genuine accident, and he was a very nice man), and in turn he had to put this action to his insurance company. This took some time and a lot of red tape and insurance company fluffing about. If I had been a public patient, I would have had the same treatment, in the same hospital, with the same surgeons, and it would have cost me nothing. I would have avoided having to sue anyone. Our taxes pay for the public system, so the most vunerable in society at least can get treatment if they need it.