Facebook has surely one of the worst traffic ever, especially if you have a generic fanpage and you're trying to drive traffic to a website. I recently got hold of a fanpage that has 650k likes and I'm trying to monetize it in every way possible. So far, I haven't seen results that would make me proud.
I know that it depends on a lot of things and there are people who are having success. I'm only judging from my own experience. Some pages might convert far better with far less likes if its audience is targeted and active, but Facebook is definitely not time-worthy as a long-term method.
I guess it's probably because of the audience type and Edgerank that I'm performing bad though.
I'll surely try the square photo thing and see how it goes.