I don't know if they have a 50-mile badge any more, but they do have a 50-mile trip requirement (2 or 3 nights) as part of the backpacking merit badge requirements.

I do remember some of JFK's fitness-in-the-schools stuff. I reluctantly participated in junior high (it was calisthenic-intensive, and I was never much of a gym rat: no scenery, no sun, and no hills.)