I would rather see a less centralized healthcare system. The Brits are in the process of decentralizing their healthcare system as they found out a highly centralized one is bad.
I would rather allow each state to create their own version. Each state can learn from the others and each state can take care of the individual needs of their own people.
You would never get all the EU nations to agree on a single system for all their countries, run by the EU. Why should all the states have a single system run by the Fed Gov?
Funny you mention that. My brother who live is Rotterdam and has EU citizenship, carries a single medical card (European Health Insurance Card). He works for IBM consulting and was in Germany when he had appendicitis attack. He was treated there in Germany, they took his insurance no questions ask. From what he has told me, single payer health care in the EU is becoming more and more a reality.