Short story about a boy who doesn't age and moves periodically from town to town

I read a short story some 25-35 years ago about a boy -- or maybe man might be a better word -- permanently stuck in the body of a pre-teen boy. He would live in a town for a few years, and then just before people would begin to notice him not aging, would hop a train and move on to another town.

I don't remember how he would explain to his new playmates and their parents why he didn't arrive with his family. Much of the story was taken up with his conversation with a mother to whom he divulged his secret. I think she wanted to adopt him and he had to explain why that wasn't possible.