If a stop light is out, can the public legally direct traffic to keep traffic moving?

You can do anything you want.

Does that mean people are legally obligated to follow your directions? No.

In my state, a stoplight that is out is treated like a 4 way stop.

I can't think of a law prohibiting so i would say yes to whether it's legal or not, but it's not very wise. One, you have no legal authority to direct traffic so the cars don't really have to listen to you and two, if an accident occurs because of something you did or by you just being there, you or now a factor in that crash and could be held responsible for it.

Yes you can do this legally, but it is safest for you not to do so. People notice police officers and in many cases are eager to do what they say (i know I dont want a ticket!), but if some random dude was directing traffic i dont know how many people would listen. like the other posters said, just treat the junction like a 4 way stop.

No. If a traffic light is out you treat the intersection as if it were a 4 way stop.

In that particular situation, drivers would treat the inoperable traffic signal as if it were a stop sign. It's a basic rule that drivers should know.