I hate to jump on an old thread but just wanted to add that this (switching the entire site to HTTPS) helped me resolve this issue too.
I upgraded from Moodle 2.5 to (both 2.6 and 2.7) and ran into this issue, immediately any time a page with a Filepicker was involved. Additionally, I was having a problem with the Site Administration links menu not loading. Switching the entire site to HTTPS ($CFG->wwwroot = "https://<your moodle site address>") helped.
Thank you Brian Carpenter for pointing this fix out.
Jim