This is due to a recent update enforcing stricter security settings. Ideally, Verizon would have to update their Java Applet to have the required permissions to run. For now, you can work around this problem by doing the following:
Go to Start > All Programs > Java > Configure Java. Hit the "Security" tab. From here, you have two options. One is more secure, one is more convenient.
More Secure:
Click on the "Edit Site List" button next to the Whitelist box. Click the Add button. Enter in the web address http://www36.verizon.com/ or whatever shows up in the error message. You do not need the full URL; just the HTTP/HTTPS domain. From here, click Add again. Then hit OK.
When you save the Java control panel settings by hitting OK to close it, you should now be able to reload the webpage you were trying to open and access the content.
More Convenient:
In the Security tab, move the Security slider to "Medium" and then hit OK. Note, this does decrease the amount of security Java provides, but this should fix the problem you're having.
Just be aware, this is only a workaround for the current version of Java 7, Update 51 for both 32-bit and 64-bit platforms. In a future update if Verizon doesn't patch this, the workaround will not work since Oracle will be blocking applets downright for not being packaged right.
The process on Mac differs slightly. Just go to System Preferenaces, and pick Java from the list of options. The control panel will open separately. This only applies if you are using Oracle's distribution of Java, and not the Apple provided version of Java.
For Linux systems, I really suggest using OpenJDK.
