We resolved this issue over a phone call. It turned out that partially complete firmware update apparently left the Egg in a peculiar state in which it was not possible to enter configuration mode. The solution was ultimately to reprogram the Egg over USB and restore it to the appropriate firmware via an over-the-air update.
The Egg is able to perform unattended updates, and it checks for them during it’s power-on self-testing sequence. While the Serial Configurator application supports invoking a over-the-air firmware update manually, it’s really only meant for unusual circumstances. As a matter of course, most users should not need to use that functionality.