I think our extremely mild January is responsible for a lot of the odd winter kill we are seeing in some of the woody plants. They started to switch into growth mode when it was so warm, but it turned cold again in February...freeze/thaw like that can be disasterous for plants.

Unfortunately, it sounds like your tree is a goner. Unless you see new growth on the brown branches, it will never recover. Conifers can not regrow from a brown, dead branch.

Does that mean that on a tree that has one brown branch, the rest will die? If not, should I cut that branch?
Thank you for your help!

A single (or few) brown branches are not life threatening to a good sized tree. I would wait a few more weeks to see if the branch puts out any new growth on the tips. If it does, leave it. If the branch is dead, you can remove it. In time, the bare hole should fill in with surrounding branches.