As some other people have mentioned below, the much lauded[1] vertically integrated health provider Kaiser Permanente started this way, as the health provider caring for the workers of an enlightened large employer, Kaiser Steel, which at the time was a massive shipbuilder for the Navy and others headquartered in California with a major presence in the SF Bay Area.
I was briefly covered by a Kaiser plan, and have a lot of old friends who are either covered, work there, or worked there, and can say people really appreciate not only the relatively low costs but also the ability to just come to one place no matter the ailment and know you'll be able to get an appointment quickly and be taken care of, and won't have to haggle to get the care you need or deal with insurance company roadblocks. To the extent there are downsides (and of course there are) they are around not being able to have your own doctor (though I do think there is an approximation of that) and overzealous cost cutting (see the kidney center scandal).
My understanding is Kaiser has not spread much beyond California due to regulatory issues.
(Interestingly, Kaiser Steel was a major setting for Atlas Shrugged.)
[1] http://www.nytimes.com/2013/03/21/business/kaiser-permanente...