Facebook app requires repeated sign-in
I had issues with the last Facebook update, the one that came out the past 2 weeks.  I uninstalled it and reinstalled it and now it works correctly.  My issue wasn't quite the same as yours, it kept redirecting to the mobile website (although I do suspect some http vs. https issue), but I would give it a try.  I know the advice seems weird because the update is like a new install, but it worked for me.