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There is no tax burden to pass along. Under a UHC system, the tax burden is directly on the tax payer in the case of medical coverage. It is never on the company. A lot of US companies don't even pay taxes through creative accounting, never mind the fact they aren't responsible for medical coverage. They get to write off any and all compensation to you as an employee as a business expense. No taxes on compensation for labor, in any form. Google only pays a 2% rate, for instance.
You end up paying for it in ANY scenario. The question is whether you want a private entity profiting off of your medical care or the government paying for it, covering the cost of its employees to do so, and that's the end of the story.
and you say my taxes wont go up, your contradicting yourself