Let me offer some thoughts from the perspective of someone who choose to drop out of 1 MBA program and later was faculty in another MBA program.
MBA programs vary greatly from school to school.
MBA programs tend to be major profit centers for schools.
With few exceptions, the coursework of MBA programs is focused on the world of publically held and multinational corporations; something totally irrelevant to a physician practice. Even those programs focused on small and emerging business tend to be focused on aspects not found in a medical practice; at least until the laws change and the corporate practice of medicine becomes legal.
One of the main goals/deliverables for any MBA program is to prepare you for and get you into your first post-graduation job. A few schools have very well respected healthcare related MBA tracks; like Penn/Wharton. Many schools offer "healthcare related" MBA programs and most of them are very lame. What do you think your first post med school job will be? What do you think your first post residency job will be?
Much of the benefit is the networking. From the right places, this networking is lifelong, well organized, extremely beneficial, etc.
Most MBA students are, or perceive themselves to be, or want to be, Wiz-kids. Much of the education isn't of much practical application, its too superficial.
Many of the post-graduation MBA jobs are glorified salesman jobs.
Much of what you learn in an MBA program is vocabulary, structure, culture, socialization, and etc. I was a business major undergrad and got a really strong foundation; the MBA courses were no where near as substantitive, and I was able to test out of several. Most of the courses aren't very difficult, although some programs can be hugely time consuming. An MBA is an essential qualification for no job that I know of, and any job that would likely be filled with an MBA would also likely be filled with a non-MBA.
In some ways, a MBA is like a post-bac special masters program for pre-med students. Its a stepping stone for some that don't have enough strength to get themselves there without it.
I absolutely would not consider doing a MBA program during med school.
If, post residency, you want a broadening experience, look into "executive" MBA programs. These tend to be much less time consuming programs with a much higher peer group level, a more practical orientation, and people who aren't focused on getting a real job because they already have one; probably someplace where you would be much more at home.