Anonymous walks the walk, but they don't talk the talk.
Even if they do shut down a few government sites, that's easy pickings. Those government sites are not well protected, not including ones with sensitive information. They're talking Facebook. Facebook is a million dollar international social media conglomerate. They have hundreds of employees, and their servers are very well protected. Anonymous is just a few thousand strong. They can DDoS the site, but other than that, it's not feasible or let alone legal for them to take down Facebook.
You're misrepresenting Anonymous, it's just a bunch of idiots looking to start stuff and pretend they something about Facebook that others don't, a juicy conspiracy. They just happen to be "a part of" Anonymous.
This is a perfect example of what I mean.