This works. However, don't use it if you install a similar one for Chrome/Chromium, and you enable synchronization of GNOME extensions in Chrome/Chromium (as per https://www.omgubuntu.co.uk/2017/04/sync-gnome-shell-extensions-google-chrome). When I used both, then GNOME integration for Firefox uninstalled GNOME extensions that I had installed in Chrome.

OK

I have the native connector chrome-gnome-shell installed, the extension works fine on Chrome [currently running 71.0.3578.98 (Official Build) (64-bit)] but fails to detect the host connector on Firefox Developer Edition [currently running 65.0b5 (64-bit)]:

"Although GNOME Shell integration extension is running, native host connector is not detected. Refer documentation for instructions about installing connector."

OK, I did get the connector to work for Debian. Add the Sid repo from Europe--as indicated on the top of the git page. Edit the /etc/apt/sources.list file from a root terminal--or with sudo (Ubuntu)--as indicated. Update the repos, and chrome-gnome-shell finally should be available. Install that then. Doing that will update to the correct python and other needed files, as well. It's my experience that the careful code-gods of Debian should agree--after a good while of cogitation and the tests of time. They'll then likely reject the Sid repo, and Debian repos will take over....

Current Debian Jessie, Gnome 3.14.1, Firefox 63.0.3 (from Mozilla), Firefox-esr 63.0.3 (native). Can't get connector to install. "python3-requests" dependency not satisfiable. "python3-requests" not in repos, either. (Got .deb file and used gdebi.) chrome-gnome-shell not in repos. gnome-shell is. Tweak tool installed. gnome-shell is in toolbar--useless--only gnome.org site usable. Which repo may prove valid, and where? A few others report this about current Debian, too.