This article's clearly written for the state of California. Did you read the article?
None of this is directly an argument against public health care, since obviously there's a benefit to universal availability in both education and health, but certainly it wouldn't be a panacea.
The same isn't true for healthcare. This bizarre system of tying your healthcare to your job means even middle and high income families don't have any peace of mind when it comes to healthcare. Heck, my wife and I are a high income couple and have expensive health insurance, and we are still completely paranoid about all the ways the insurers could find to screw us over, especially now that we're about to have a baby. Every time my wife goes to a pre-natal visit, she ends up fighting with the insurer about how something was coded, etc.
Sorry that this sounds like an ad, but we don't all live in fear of our health plans. (I dislike the term "insurer", since what they sell stopped behaving like insurance a long time ago.)
Even with the "high income" surcharge here for health-care, it's less than $1000 a year for an individual. Since it's deducted from your income along with regular payroll taxes, you don't even have to pay for dependents. They're covered under their own plan which is basically free until they start earning and paying deductions of their own.
In a start-up environment, $12,000 a year is not a lease on a Mercedes, it's the difference between your business floundering in obscurity and affording a few key networking trips, or the difference between living in a bedbug infested hellhole or having a decent apartment.
There are private insurance plans for exceptional circumstances, but these often over-lap with other policies to such a degree they're basically a luxury offered by companies to entice workers. The only real perk to them is the dental and optical coverage that isn't covered by the standard health-care system.
There's also regulated costs, which makes it cheaper across the board. Per-capita spending is actually only about 50% what it is in the USA (http://en.wikipedia.org/wiki/List_of_countries_by_total_heal...).
Factoring in that, the net cost might be subsidized by other taxes by at most ~$2000 more a year. Part of this is paid by the employer on behalf of the employee and isn't listed as a deduction. The rest is subsidies from the federal level of government.
The thing that makes this more affordable for people on limited incomes is how it's tiered, not a fixed price for everyone. If you're an entrepreneur barely making an income, you don't pay much.
I would point out that these kinds of things are about trade offs. As a society we are much worse off without a national education offering, even if it has significant problems to overcome. At this point in time, private health insurance tied to employment have clearly failed as a way for US society to have decent health care, as we spend more for lower quality service than most other nations in our economic class. Universal health care programs are not so much about a panacea, but something that doesn't suck as bad.