After some digging into this myself, I've discovered it is a MIME type issue. Firefox (or Chrome) on Linux won't play SWF files if it thinks the type is application/vnd.adobe.flash.movie.
Edit /usr/share/mime/packages/freedesktop.org.xml as root and replace the following:
And then run:
I found I needed to restart Firefox after. Here's a script very close to one from http://ubuntuforums.org/showthread.php?t=2218732&page=2 that performs these steps (run it with sudo), but I will warn you -- I noticed that different installs seem to have different spacing, which can trip up the sed script, which is what happened to me.
echo "Update file freedesktop.org.xml...";
sed -e "s/<mime-type type=\"application\/vnd.adobe.flash.movie\">/<mime-type type=\"application\/x-shockwave-flash\">/g" /usr/share/mime/packages/freedesktop.org.xml > /usr/share/mime/packages/freedesktop.org.xml.new
mv /usr/share/mime/packages/freedesktop.org.xml /usr/share/mime/packages/freedesktop.org.xml.original
mv /usr/share/mime/packages/freedesktop.org.xml.new /usr/share/mime/packages/freedesktop.org.xml
echo "File updated successfully!";
echo "Update mime database...";
echo "Mime database updated successfully! ALL DONE!";