I have the solution (at least for me). Hopefully this will help someone else...
I was watching a video podcast from MacBreak Weekly and they were discussing changes in Mojave and how some apps now need special settings to work. This led me to a fix for my 1Password application that was misbehaving and I wondered if I could find a solution to my EyeTV problem in the same general area.
So I went off to System Preferences > Security & Privacy and clicked on the Privacy tab. I scrolled down to the Camera item in the sidebar and saw EyeTV listed there but unchecked. This is to “Allow apps below to access your camera”.
Now just why EyeTV would need to access my camera is a mystery to me but as a test, I unlocked the padlock, then put a checkmark beside the EyeTV item and tried to launch the EyeTV application. It came up properly and I could watch my TV shows. Then I quit EyeTV, went back and unchecked this box and tried EyeTV again and it crashed as before.
So I would love to know why EyeTV needs to access my camera, but at least I have found the solution.