@lan15
In my humble opinion, for Health Management, hands-down HSPH. It is the most suitable one since you can take classes at HBS, under world-class faculty (although you might get second dibs on registering for classes as a non-MBA candidate, and I am so sure of that) and build one of the best connections and network. You can also cross-register at MIT and take some amazing classes. Additionally, if a start-up is something you are looking at, nothing to beat HSPH with the Harvard Innovation Lab. I also believe that with "fixed semester fees" you can take as many classes as you want and load up, if you are into the nerdy thingamajig.
If you want "all things public health" then JHU is the best. The choice of classes you can take is phenomenal and you will never run out of subjects to learn, and the label global now. But the business school is not up there with the top ones.
I am not sure about Columbia, but I would think that if you want just the ivy thingy (which you get from HSPH, anyway) and live in NYC, then it matters but otherwise, not much compared the other two.