Walk me through a typical set of steps that leads to the cropping problem.
Is it something like this?: You download a new image to FAA, and your Facebook personal ap is set to post it on Facebook?
My personal Facebook automatic posting is not enabled, ... so I don't know how this would work. But my shopping cart ap on my Facebook fan page seems to show my images in correct proportions - the SQUARE ones, that is. I have not really looked at the non-square ones.
Here is a link that describes optimal sizing for posting to the various Facebook locations:
http://havecamerawilltravel.com/photographer/images-photos-facebook-sizes-dimensions-types
I would imagine that all images are somehow crammed into these specs, once they are uploaded. Facebook specs dictate appearance, and so unless you are willing to monitor and size every single image of yours that gets posted to Facebook, it looks like you get what you get, ... at least as I seem to understand things at present.