The Java FX website says that it is available only for Windows and Mac OS platforms. What does this mean to a web developer with Linuy as the OS. I mean if I use Java FX to develop my screens and when I deploy them to an App server in Linux, would it break?
JavaFX is a client-side technolgy. If by "deploy them to an app server" you mean you'd be using Linux to serve them to a client, then there should be no problem.
I also think it should be possible to run them on Linux - JavaFX is written in Java, after all (for the most part). I think JavaFX 1.1 (and maybe 1.0) added some multimedia codecs, though, which may not be available on Linux. But the basic stuff should work fine.