As a request to our client we changed to HTTPS on our server for their Moodle 2.7.1+ instance. I configured the site, but after the change the file picker stopped working.
I have found that Chrome console shows the following:
Failed to load resource: the server responded with a status of 404 (Not Found) https://c328740.ssl.cf1.rackcdn.com/mathjax/2.3-latest/MathJax.js?delayStartupUntil=configured
Uncaught ReferenceError: MathJax is not defined yui_combo.php?m/1408696391/core/event/event-debug.js&m/1408696391/filter_mathjaxloader/loader/loade…:151
It was definitely true. I tried the link with HTTP and it was fine.
So my solution was to delete the equation from the Toolbar config at Site administration / Plugins / Text editors / Atto HTML editor / Atto toolbar settings.