I never said the costs would be lower. In fact, the cost issue is the only real argument I see against it.
Why not offer incentives (tax breaks, for example) to coerce the system into the right state? For example, if you go to medical school or nursing school and graduate in the top 50% of your class (from any accredited school), you would be automatically reimbursed for the entirety of your tuition by the government. This would create an incentive for more students to become doctors and nurses, which is sorely needed today; at the same time, it reduces the barrier to entry (the cost), creates motivation to do well, and also keeps the price of service (what the doctor charges you) down as well (since they're not having to pay back $200k in student loans).
At the same time, the system should require everyone to get certain tests on a regular basis (e.g. a yearly physical). If you don't take care of yourself at some kind of baseline level (the doctor should be able to tell from your charts), then you will be charged a "surplus" fee since you will be requiring more medical care down the road (e.g. heart disease due to lack of exercise). This would motivate people to get in shape, or at least they'd be paying for the extra costs of their own care. So, essentially, it's free as long as you play your own little part in the system.
On another note, all the people that I've heard argue against UH have all said that they don't want to pay for someone else's healthcare. While I understand where they're coming from (I hold mostly libertarian political views), you have to remember that the 'healthier' our society becomes (due to UH, or simply lifestyle changes, etc.), the "cost" per person is reduced. This is because healthier people are more able to contribute to society, rather than unhealthy people who must survive on the contributions of others.