I have had hackberry get that green stain, too. It was freshly cut and I just figured it was a feature of the wood and not something I should worry about. In the case of my staves, it tended to be right where there had been some damage to bark when it was being cut. Since it happened so fast, I doubted it was a fungus, probably some response the wood has to injury.
Gumboman, I think hackberry is every bit as good as osage. I have never had a hackberry bow break. I have screwed them up with bad hinges and crappy tillering, but never had one break. They have taken set from bad tillering, they have had poor cast from poor tiller, but never have I had one let go.
I think the wood is more "rubbery" than osage. Well cured osage parts in front of a sharp edged tool easily and cleanly, it is a pleasure to run long, clean curls with my well sharpened spoke shave. Hackberry works like an old truck tire, rubbery and resistant to parting easily. It takes more effort to pull curls with a spoke shave, no matter how well stropped and hair popping sharp the blade is.
I think hackberry is a prettier wood when finished. It is a ring diffuse wood like maple, versus ring porous woods like osage and oak. Hackberry can be a pain in the sitting muscles when it comes to taking stain if you sand it out too fine (don't go past 180 grit if you want to stain it darkly.) But you can take advantage of that to create differentially stained finishes. For example, I once sanded the back of a hackberry bow to no more than 150 grit, taking care to follow the grain religiously. The belly was sanded with a random orbital sander all the way down to crocus cloth. The back stained nice and dark and the belly barely took any color at all. I loaned that bow out at a Rendezvous and never saw it again.
Folks often mention the smell of hackberry. To me it smells like hot buttered popcorn or corn tortillas cooking. I believe the wood has a fairly large amount of sugary compounds in it, and because of this the wood will toast nicely with careful application of a heat gun. It handles heat treating well, but will fool you and spring back if you do not get it well and truly heated deeply into the limb. You really need to take your time and heat slowly and thoroughly!