In theory you can postpone your residency indefinitely (after all, no one says you HAVE to practice medicine just because you have an MD). There is no limit as to how long after you can enter the Match (I believe), however, I don't recall if the USMLE step III needs to be taken within a certain time after graduation (I don't think so).
I think however your question is what are the practical aspects of not going to residency directly out of medical school.
In general, people should probably go directly to residency after medical school since they are as current as possible on clinical knowledge. One exception that is not uncommon is that in very competitive specialties people have been known to take a year to do research to beef up their CV. (IMHO this is not necessary for >95% of people).
As far as doing non-medical things, well it varies, but by and large I think program directors would see this as a negative since your clinical knowledge is getting stale, so the longer the break the more it will impact your ability to get a good residency. (again IMO). Doing MPH's after finishing the MD are kind of a wash, probably the improvement in research skills barely balances out the extra time so there is no negative impact. An MBA on the other hand, I think is probably not going to be viewed positively. There is a GOOD reason why the combined MD/MBA programs have you finish out with at least 50% clinical experiences in your final year, 99% of the time.
I would advise your friend to wait and get her MBA later in her career unless she anticipates wanting to not practice clinical medicine. She will get a lot more out of the MBA with some more experience and she will have many more options.
The short answer is thus: you CAN postpone residency as long as you want, but anything over 1 year of non-medical (i.e. non-research, non-medical volunteer) activities is going to be looked on with suspicion.
