Can git undo a checkout of unstaged files

If you are using a "professional" IDE chances are good that you can restore files from a local History. In Rubymine for example you can right click files and watch a history of changes independent from the git changes, saved me a few times now ^^

I believe if a file is modified but not yet added (staged), it is purely "private".
Meaning it cannot be restored by GIT if overwritten with the index or the HEAD version (unless you have a copy of your current work somewhere).

A "private" content is one only visible in your current directory, but not registered in any way in Git.

Note: As explained in other answers, you can recover your changes if you use an IDE (with local history) or have an open editor (ctrl+Z).

If you are working in an editor like Sublime Text, and have file in question still open, you can press ctrl+z, and it will return to the state it had before git checkout.