The Boy Scouts Of America are looking to recruit the future Ron Swanson's of the world, but you don't need a beard to join, you just need the desire to "be one with the wild" (their campaign slogan). Only time will tell whether their ads, featuring kids with epic beards, will increase their numbers or scare the wits out of potential recruits!
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Anyone have ideas for other international organizations that accept everyone and don't base their creed on belief in the supernatural?
Unfortunately I do not. I will look around and if I find one will post it here.
Looks like http://navigatorsusa.ning.com/ may be what you are looking for. Looks like it was started by an ex-scout leader.
If your local Scout troops are on the Jesus Wagon, you might try 4-H clubs at the high school level for an alternative. In my opinion, that's far more valuable anyway. Scouts bored the crap out of me, and it didn't give me any advantage in the military. I learned more about survival and outdoor skills in boot camp and infantry training school than I ever did in the Scouts.