You should be able to run the updater from the desktop shortcut without any problem. The updater's actually a Java program installed on your computer, not a browser-contained applet, so the blocking of the Java plug-in by Firefox and others doesn't affect it. The only thing that might be affected is initially installing the updater via the link on the Web page, and even that I don't think is affected because it's a JNLP link and not an applet (applets are handled by the browser plug-in, JNLP links are handed off to an external program that's part of the JRE to download and install).
The complete fix is to download Java 7 update 11 (it went live yesterday) from
http://www.java.com/en/download/manual.jsp and install it after removing prior versions of Java including Java 6 (there's no patch for Java 6 planned, so it'll be permanently vulnerable).