Hey, I know Flash works using Facebook, and I think I've seen some Java stuff on there too, but does anyone know if the Unity Player can be used in Facebook?
The Unity Player can be used on facebook.
One relevant thread on integration: http://forum.unity3d.com/threads/7812-Unity3D-meet-Facebook!
And one very popular game: https://www.facebook.com/uberstrike
Here's a thread on Unity Answers that also has some more details: http://answers.unity3d.com/questions/82/how-can-i-make-my-unity-application-work-inside-facebook
This is kind of a nonsensical question based on a poor understanding of how Facebook apps work. Facebook doesn't really have anything to do with what technology exactly you are using; they simply embed your app as an iframe on their site. You build your app using whatever technology you want on your own server, and then simply tell Facebook what URL to point to.
So basically anything that can be done in a web browser can by done as a Facebook app. Now there are certain technologies that get used more often for Facebook apps, but those trends are due to reasons that have little to do with Facebook, like the fact that most browsers have the Flash plugin.
(I'm not criticizing you btw; until recently I didn't understand how Facebook apps work either, and thought they were hosted on Facebook's servers or something.)