It is possible that you have an XPI addon that is preventing you from doing this function. If this is the case then start Firefox in safe mode using the icon in the start menu this will disable all XPI's and try again.
It may also be that your Java Plugin is disabled as shown below you will not always see the Click here to manage your plugins.
You will need to find the Java (TM) Platform SE Plugin this the the latest version v22.214.171.124 at the time of posting
This should work however;
If you still can't get it to work then use the IE tab
XPI you will have to logon to your account again this way you can switch to the IE shell,
You will then have to accept the software install from Facebook,Inc click more options and then click Always install, you only need do this once.
This will then allow you to upload using the Java the simple uploader option is a bearable alternative but you can't rotate jpegs.
If your browser brings up a warning like the one above this is because the software from Facebook is unsigned just press OK.
If you still have problems I suggest clearing out your older versions of Java Runtime using the excellent JavaRa v1.15
at the time of posting this is the latest stable version.
You can also try the New Photo Uploader Prototype from Facebook if all else fails this has been written by a Facebook engineer due to the problems that the Java Uploader gives many people.
Check out my tip HERE
on how to best do this by donwloading the uploader plugin first.