I don't know if you are aware, but the H2 is a Silverado chassis with a "Hummer" body. The H3 is a Colorado with a Hummer body. The H1 is the only one that is not based on a Chevy platform. It is based on the AM General HMMV Military vehicle.
So, if you need parts for the engine, trans, suspension etc. you are in good shape(H2/H3). If you need body parts, have fun and good luck.
Off road they are no better or worse than the truck they are made from. They will have a little better approach and departure angles though, very short overhangs front and rear.
Aftermarket support is going to be slim due to Hummers no longer being made. There is no H2/H3 diesel.
In the end they could work, but I would suggest that if you go that route, get a good deal on one I don't see them having a good resale value anytime soon. And be ready to get creative.
As for the H1, its kind of cool in a way. A real HMMV would be cooler, but they are slow, underpowered, and WAY too wide for a lot of trails.