When we first moved to our present home, which is surrounded by woods, I began to notice white branches which seemed to appear in the dead of winter every year. For several years, I could not figure what caused it. They appeared to be completely stripped of their bark - and they were. I finally discovered the cause one winter when I observed a squirrel out over the deck strip a long piece of bark off a branch of a sweet gum. It started at the far end and stripped it all the way back to the trunk in one long piece, then took quite a bit of time to ball it all up and make sure there were no hanging pieces that might trip it up as it jumped through the trees. Once it was secured, the squirrel climbed up to it's nest in a nearby tree.
Note how this branch has been stripped of bark.
After watching this same thing over time, I began to realize that squirrels don't pick just any branch, but evaluate them based on how many smaller branches are attached. (Notice how the branch above has no secondary branches.) The fewer the branches, the easier it is to strip them back. And these are the ones they appear to choose. It raised my estimation of their abilities.
You would think bark would not make very soft nesting material, but an abandoned nest blew down out of a tree onto the lawn a couple of years ago and I had a chance to examine it. I'm not sure what happens to the outer layer, the woody part of the bark, but they appear to use only the cambium, or inner tissue which is much softer and makes an ideal bedding for new babies.