I think the fact that Peter Pan lives in Neverland is an indicator that, at least to Peter and the lost boys, childhood is a journey and not a race. And for those boys, it is a journey that will not end, because in Neverland, they won't grow up.
I've heard the phrase "when will he/she grow up" numerous times in my life referring to any number of people. I also heard the "act your age" comment before. I'll admit, I even use it on my own kids. "You're six years old, start acting like it" or something like that. Those kinds of comments are all geared toward the attitude of having people grow up as fast as possible. That's a race. Get in, get out, and do it faster than everybody else.
Peter Pan on the other hand joyously sings "I won't grow up" because he is not a part of that race. To him, every day as a kid is meant to be enjoyed and full of play and adventure. He wants to have Indian wars and annoy Hook and eat too much sugar with his lost boys. That's part of his childhood journey. Carpe diem. He wants to live those moments to the fullest. He doesn't want to win the growing up race and leave all his adventures behind.