My son is not completely non-verbal, he can communicate his needs and wants in either full sentences, or phrases, but also has ADHD. We first started taking him to MyGym (a gym for kids with different locations) with his ABA aide and very recently began private gymnastics lessons with an ABA trained coach. He loves it and is more inclined to focus and follow directions, because it is a desired activity.

Here is a link to the MyGym website to look up locations near you.
http://www.mygym.com/locations

We had an autism bowling league and my daughter loved it. Some of the parents did have issues with kids running, but most really enjoyed it.

Martial arts?

What about a bike ride or walk through a trail or a picnic in the park. Try feeding ducks, children's library we do crafts a sprinkler out back.

my daughter also loves playing in the sand. She also loves her sidewalk chalk, and has become very interested in painting. I bought her huge I mean huge paper and she paints with obviously washable and it is the second longest activity she will do other than splash parks