Ok, I circled back around to this and figured it out. It was staring me right in the face.
The following function in CRM/Myextension/BAO/Query/Soundex.php needs changed. There needs to be a hidden field added to the search form for each advanced search pane. It was so simple in CiviHR example that I look right over it and went down the rabbit hole.
As Laryn notes, it should show up in the Custom Fields section of Advanced Search anyway providing the field is marked as searchable.
You can search by callsign in the Quicksearch box by going to Administer > Customize Data and Screens > Search Preferences and ticking the callsign field in the Quicksearch options box.