I doubt whether the plugin requires the latest version of Java but Java itself. With the many security problems Java has seen access the years it probably wants you to update to plug any holes.
You can install the webplayer plugin manually or maybe you should uninstall Java and install either 32 or 64 bit Java and try again? I don't believe I have both flavours running on my Windoze laptop.
Does this affect other PCs? If not, then it is probably a local issue as opposed to a plugin problem.