Except, at least here (Canada) we haven't cut out insurance completely. Not everything is covered under provincial healthcare (e.g. dentists and prescription drugs for most people). So, most of us still have insurance.
Mine is through work and I don't, outright, pay anything for it. It covers 80% of most things (e.g. drugs, eye doctor, dentist, physio, massage). Some things have yearly caps (or 24 month caps) after which it doesn't pay anything. Other things, like drugs, have a cap ($1000) after which it pays 100%. My deductible is $25/year (one for drugs/health care and another for dentist - so a total of $50). I also have about $850 in a spending account that I can use to pay the extra 20% or things over the cap.
No, I say that I don't pay anything BUT - it is a taxable benefit (about $17 per month). Also, they way our work does it, we all get $X in credits (the amount depends if you are single, a couple or a family) and you "buy" your options. So, you can choose from 60%, 80% or 100% coverage (one option for dentist and another for healthcare - so you can do 60% in one and 100% in the other). These credits are also used to buy life insurance (above the standard 2 times your salary), LTD (above the standard 50%), and accidental death. Any extra credits can go into an RRSP (retirement savings plan), health care spending account, or be taken as cash. You could also pay in if you end up choosing options that come to more than your credits. So, by taking more than the bare minimum, I am losing money that I could have received; I think my options were about $700 per year - so it "costs" me $700 plus the taxable benefit.
Other companies do things differently (and there are people who don't have company plans and purchase their own insurance). But, the main point is, a single payer health care system does not remove all need for health insurance.