An interesting study. No good, easy answers (and I would not expect any), but I thought this is probably the biggest answer (emphasis mine):
Is it health care? This is a popular theory because all British citizens have access to a government-run health care system that encourages preventive care. Despite that striking difference, however, researchers hesitate to cite it as the explanation. After all, Americans spend nearly twice as much as the British do on health care, and many of the wealthiest Americans in the study, who presumably have insurance, are sicker than are the poorest Brits.
One thing about the American health care system (and I use the term system lightly here) there is not a lot of emphasis on preventative care. Few, if any, insurance companies work out deals for discounted gym memberships, weight loss programs, stop smoking programs, etc. I think insurance companies are starting to wake up to it, but we still have a LONG way to go. I am not even asking for the insurance company to make the programs available for free, just work out some kind of discount.
The stress pointed to in the article is very real. Brits I believe get 3, maybe 4, weeks of government mandated vacation a year. In the USA we get 0, although most companies give 2 weeks after 1 or 2 years of service. Even then most people are encouraged (and in some cases ordered) to not take their allotted vacation time. Even for those who do take their vacation, the vacation is as stressful as their job. A different kind of stress is still stress.
As I understand vacation in Europe, people will just take a week (or more) and get a house in the country and just kind of do nothing. No forced marches through Disney World, no "are we there yet" road trips. Just some time to relax and unplug from the world.