I have a git repository (and working directory) that is stored in my Dropbox so I can move back and forth between computers without having to commit or stash (read: without any
I've gotten this exact problem under the same circumstances.
To stop the files showing in git status you can set git to ignore file mode changes:
git status
git config core.filemode false
This change is local to the git repository and will be synced along with the files.