Competition in a free market... boo. That's all I got on this one. Greg said it all.
As for the last part... the closest example to give you is auto insurance. One of the factors based on your auto insurance rate is the number (app. percentage) of uninsured drivers in your state. Basically, the more uninsured drivers in your state, the more you pay.
Health care works similarly. Prices do go up with the number of uninsured patients. Hospitals have to charge more to make up for the free rides they give in the ER. If prices for procedures go up, your insurance pays more and you pay more. Basically the theory behind it is that if everyone is covered and hospitals are guaranteed to get paid then the costs go down. Whether this will work is hard to tell, hospital administration is a hugely complex business.