Sort of? Past a certain income level ($97 kAUD), you pay an additional levy (1 - 2%) on your taxes for the public system (which goes into supporting it), but otherwise can access it for relatively low cost/free if you’re a citizen, in an insurance-like system (if you go to a place that doesn’t do it automatically, but still qualifies, you can get a rebate).
You can avoid paying the tax if you opt for private health insurance, and don’t use the public system altogether.
Neither of them are tied to employment, although some employers do offer health insurance as an extra addition in pay package bundles, like they do with cars and things.