I have suffered with this issue for 4 days now, where even with a active internet connection the facebook App for Android will report “no internet connection” I finally solved it, first this will require that you have your device rooted, and a copy of Titanium Backup (not sure if you need pro or not, I have pro)
First uninstall Facebook.
Load up “adb shell”
I ran from the /data directory the command rm `find . | grep facebook`
re-install the Facebook App
open up Titanium backup
Find facebook in the list and select it
from the dialog box that comes up select Wipe Data
For me that fixed Facebook for Android. But Facebook needs to fix the root problem what ever it is, as not all users have Root Access (I am lucky I do as I am a geek) A simple uninstall/reinstall should fix it.