Attach Screenshot no longer works on OS X after Java Update

Steps to Reproduce:
After updating OS X (Lion) to the latest Java available through the Software Update, you get:

java version "1.6.0_37"
Java(TM) SE Runtime Environment (build 1.6.0_37-b06-434-11M3909)
Java HotSpot(TM) 64-Bit Server VM (build 20.12-b01-434, mixed mode)

When you open an issue and go to the Attach Screenshot option, it asks you to Update the Java Applet and sends you to http://www.java.com/en/download/mac_download.jsp?locale=en if you agree to install. After installing it if you check Firefox Add-ons the plugin is:

When you go once again to the Attach Screenshot option, the Applet loads but nothing happens when you click on the Paste button.

Expected Results:
Screenshots should be pasted and added to JIRA Issues.

Actual Results:
Nothing happens, no errors in the logs or in the screen.

Workaround:

Revert the Java Applet update as described in this Apple KB: http://support.apple.com/kb/HT5559?viewlocale=en_US&locale=en_US