In the US if you quit your job you can buy the same health insurance plan you had, for up to 18 months ("COBRA"). It's not going to be the cheapest thing in the world, but if you're talking about taking a year off, you're probably not on the verge of going broke. (It looks like it'd be far less than what I pay for rent, for example.)
Another option could be to take a part-time job. A fair number of stores seem to offer health insurance for any employee working 20 hours a week. Of course, that may or may not be compatible with one's idea of a "break".
When I quit my job last year to join a start-up the cost to continue my COBRA benefits would have been $700/month (and I'm a healthy guy in my 20s). Fortunately, I'm in MA with its private health insurance market, and I was able to find a plan for $200/month. I still had to go without health insurance at all for a month, though, to make myself become inelegible for COBRA and therefore eligible for the other plan.
"Not the cheapest thing in the world" is quite an understatement.
From what I'd found (also for a young single person), it would have been more like $500/month, but even at $700, that's still cheaper than rent at any apartment I've seen since I was a college student.
And remember the context: this is someone who wants to take a year off with no work at all, and so is voluntarily giving up tens of thousands of dollars, in exchange for his own time. Yes, there are almost certainly cheaper health insurance options, depending on how much effort you want to put into the research, but COBRA is the upper bound, and takes practically no time to sign up for, which, again, was the whole point of the exercise.
Have you checked these claims in the "real world"?
Costs have been rising substantially each and every year. The number of employers willing to give even full time employees insurance has been shrinking and the cost of your employer's health insurance plan willing be quite high considering you don't have your employers tax incentives.
It's not hard to find names of employers who offer healthcare even for half-time workers, using any "search engine". I'm sure there are also companies who are dropping their health insurance plans, but that doesn't mean that all companies are.
Another option could be to take a part-time job. A fair number of stores seem to offer health insurance for any employee working 20 hours a week. Of course, that may or may not be compatible with one's idea of a "break".