Thanks for all your efforts FurryNutz. Having been through quite a few Java updates and MyDlink updates over the past year, I'm convinced this is simply a case of Dlink not working with the newest version of Java. Changing one's browser settings probably isn't going to help, and working with Dlink is probably what will fix it. Thanks for working with them on this.
My cameras work fine with other software, so I know they are all right. I can access the cameras directly through their IP addresses as well. I'm convinced the Java update caused this latest problem because access through Dlink worked, suggested the update, and then it didn't work immediately after the update. All during the same session.