I forgot to not put a number of Eclipse settings files into my git repo. I removed them from the current branch. However, I now cannot change, via Egit, to any branches that still track them.
I get the "checkout conflicts" dialog and none of the three options provided do anything. I can't commit or stash them because as far as git is concerned, they don't exist and resetting the branch does nothing.
I ended up fixing the issue by using a different git instance and then reimporting the project from the repo back into Eclipse.