Boy Scouts 50 miler?

I am curious if these are Boy Scout 50 miler kerchiefs, also what era are the badges. I found these in a box of American Flyer Trains from the 50's or 60's. I posted in a group on facebook and got a variety of comments. The one that says 50 miler actually looks like a woman to me, even though I was told it is Boy Scouts.