You didn't specify what type of spruce...so...these are some general guidelines. Most Spruce hardiness zones range between 2-7. So they can with stand the cold. You could heel them in with plenty of mulch and perhaps a wind break with some burlap and stakes or even a couple of hay bales. Not knowing what your cold frame is made of and how much sunlight is filtered would be a tough one to answer. Spruce like light even in a dormant state.
I would be a bit leery about your spruce either way. Collection of this species in summer is risky. Some general guidelines
here to help you.