A quick anecdote: my friend and I here were laughing because in both cases our pregnant wives went to the doctor about a small concern and were put in the hospital for a week for testing. But I thought about it later, and that wouldn't happen in America. You're churned out as fast as possible. The great side effect, and I think you'll see this on infant mortality rates as well, despite what you've said, is that if health care is free, people use it earlier than the crisis point. And that's cheaper and better for everyone.
Or rather, it saves money for the individual who has something worthy of being prevented. For that person, it's much easier to find something and/or prevent it early rather than wait until it gets serious. This is why prevention seems like such a common sense thing.
For society at large, however, testing and servicing all of the other people -- the people who never will get the thing you're trying to prevent -- ends up spending more than the treating person who got it later.
It's one of those things that sounds good when you say it, but the statistics don't back it up.
But it's not more expensive. Most European countries spend much less on health care, but much more on prevention. It's very effective, and at the end of the day, it's really what medicine is about.
http://www.medscape.com/viewarticle/540199
...After all the consequences are accounted for, very few services of any kind save more money than the cost incurred. Childhood vaccines are among the rare examples that actually are cost-saving...
Of course, there's more to universal health care than just saving money. Once you politicize it, it's more about doing and saying things to keep those votes coming in.