The photo suggests the outer shell was killed while the protected inner growth survived. A common pattern from wind desiccation and freeze damage. I would trim out the dead material gradually rather than cutting it all the way to the ground immediately.
Scratch-test branches first. Lightly scrape the bark with a fingernail or small knife. Green underneath = alive. Brown/tan and dry = dead. Cut back obviously dead branches to where you find green/live tissue. If entire stems are dead, remove them at the base.
Replacement is often easier and gives a better-looking result sooner. This shrub looks heavily damaged, so recovery could take several years even if it survives.
My Boxwood is Wintergreen Boxwood. It is said to be the hardiest of Boxwood for colder weather. Cold Hardiness -20 to -30 degrees F. But non-hardy in zones 1-3. It is slow growing though.
I bought a couple of Wintergreen Boxwood locally here and one wasn't doing very well. So I ordered a Wintergreen Boxwood off of Amazon. And the one I bought from Amazon is doing much better than the one I still have that was purchased locally.
