There's an anomaly between some of the newer routers and Android phones. The speed is generally a bit on the slow side (not much, maybe 10%), but the Facebook app just dies (even though going to the Facebook website works fine).
I solved it by buying a $35 Netgear N300 Router, knowing I could return it if it didn't solve the problem. (Linksys and D-Link have the problem, so I figured I'd get as far from them as possible, and still stay with quality.) It works out of the box. I just set the SSID and password the same as the Linksys that didn't work (and was pretty new), turned it on and waited for the phone to connect to what it thought was the old router. All the other devices in the house were connected by the time I looked at them.
(BTW, if you set up a guest account on your router and connect to that, the Facebook app will probably work, but who wants to use an unsecured wifi connection?)
I haven't figured out what's causing the problem, and I probably won't waste the effort to figure it out. And Linksys doesn't seem to care.