As several of my health care brethren and sistren have already mentioned, yes, gallows humor abounds as a coping mechanism in what can be emotionally draining if not outright soul-lacerating work. As the old saying goes, if we weren't all crazy we would go insane.
However, it's generally understood that you keep a tight lid on that stuff. I don't think Deena using the term "nut job" was a big deal in itself, but I do think using it in a classroom was pretty stupid.
Deena made an error of judgment and got called on it. From the information given it seems the wisest course for her would have been to stand up and own her error, learn from it, and drop it. Instead she seems to want some sort of absolution for it so she keeps escalating the situation by pushing to find someone who will tell her she didn't do anything so very bad, it's simply a case of this professor being a big old meany poopy-head. Now she wants to take it to his bosses? Really? She needs to sit down and think about what this is really about. We all make mistakes and accepting that we do and taking responsibility is part of our growth as human beings. This is a pretty minor one as things go.