This bush is about 4 feet tall, and is growing in the shade. It's berries are hard like barberries and we found them in June. They may have started in May. We didn't see the flowers. The berries seem always to be in pairs. Any ideas?

Maybe this one: Lonicera canadensis (Fly Honeysuckle)

Definitely a Honeysuckle species - Lonicera sp. - with the pposite arrangement of simple foliage on this woody shrub

Definitely NOT a Barberry of any sort; they will have alternate arrangement.

I just found this link. You do have the American Fly Honeysuckle, which is awesome! Last year I bought three little saplings by mail from Reeceville Ridge Nursery in central Wisconsin. One died, but I have the other two doing well in pots until they get big. The smaller one is bearing the tiny creamy flowers in pairs that will produce the light red berries in pairs. For me it is a rare plant, but up in the Northwoods it may still be common in places.