This thread is over four years old, so I doubt any of the original posters will be here, but here are my somewhat uninformed thoughts.
If you want to get into healthcare consulting in a major firm like McKenzie or Bain, my feeling is you're better off with an MBA in healthcare management from a TOP business school. By top, I mean top 10 in the country. I'm talking like the Health Industry Management Program at Northwestern (Kellogg). Realistically though, the key to getting into one of these major firms, if there is one, is an MBA from a top business school like Harvard, Columbia, Stanford, UChicago, etc. They love prestige and "brand name" schools at those firms, and a lot of the people working at those firms are from these business schools. The MBA is also just much more recognized in those firms than an MPH or MHA is.
If you're set on an MPH for this, Harvard and Columbia would be your best bet, as the affiliation these firms have with those business schools would be helpful. After that, I would argue U Michigan is a good choice as they have a strong business school, powerful alumni network, and a great reputation in healthcare management. UC Berkeley, given its strong business school, would probably be a good choice as well. Beyond that, it gets a little fuzzier because these big firms inordinately care about idiotic stuff like Ivy League affiliation and as such honestly might value a Yale MPH over one from U Minnesota, despite the fact that U Minnesota is #2 in the country for healthcare administration.
Keep in mind, this is just for major consulting firms like McKenzie. I'm sure for healthcare consulting in general, it's not so intense, but my feeling for those big firms is that 1) MBA from a top business school is your real ticket in 2) They care way too much about name recognition and prestige, and 3) They probably won't value an MPH or MHA as highly as an MBA. Just my thoughts, based on what friends have told me when they worked for major consulting firms.
Edit: Just noticed you're already an MD. Given that, I would definitely say an MBA from a top business school would be more beneficial to getting into a major consulting firm than an MPH. You already have the medical knowledge with your MD training.
Another caveat, everyone I know who has been involved with major firm consulting hated it and burnt out after a few years...