Thank you for the post and I apologize for the delay.
"We recently updated our server from FileMaker Server 12 to 13."
The port used for steaming Interactive Content changed with FileMaker Server 13 from the port used for FileMaker Server 12.
FileMaker Server 12 used port 16000 "to progressively download content for FileMaker Pro/Web clients."
FileMaker Server 13 uses ports 80 and 443 for "Progressive downloading of container data to all clients" and for "inserting data in externally stored container fields from all clients."
As a test, eliminate the network by installing FileMaker Pro 13 on the FileMaker Server 13 computer. Create a New Record > Right-click on the container field > Insert.
If the contextual menu is available on the server, and the content displays as interactive after inserting the image when viewed from the local FileMaker Pro client, then one or both ports are most likely blocked for the external clients.
Please let me know if any of the above requires additional clarification.