Feel free to close this-
After a little bit of monkeying with each setting under [ -- Add Other Field -- ] in advanced search, I found a combination that works if anyone else was looking to do something like this. We are not using osTicket in the traditional sense and needed something to show overdue tickets with a specific (or without a specific) status.
Here's some screen captures of how we got it working.
Advanced Search -> Add Other Field -> Flags
Set 'Flags' to Overdue
Set status of 'IS NOT' for every status you don't want to see (for some reason, IS and a specific status does not work in my environment).