Well, if UHC decides to negotiate hard and fails, then people will leave UHC and move to another insurer which is willing to pay those insane prices.

If country X (with single payer plans) does the same thing, the only choice the pharmaceutical company has is to not be covered by insurance, which mean they won't sell their drug. Also the state can just pass laws to make the pharmaceutical company life's miserable. UHC can't do that.