I've heard that chickens will take away all of the cover crops, and was wondering if there's any type of cover that will survive chickens.
At least within permaculture chickens are employed to get rid of any weeds and other types of hardy plants. I dare say that "all cover crops" really means all cover crops. Don't let chickens near your cover crops.
If you still want to use chickens in your forest make a movable chicken coop (a chicken tractor) of a decent (big) size and see what they do to your crops in a couple of hours. To minimize the damage, I would probably try something like potato (tubers) or root vegetables like beets or turnip first.
Even if they turn out to damage your crop, you can still move the tractor over unused areas, which will help keep unused parts of your forest clean (and prepare the soil for future planting).