I really didn't follow exactly what you're were telling me to try (I'm pretty new to Designer) but you got me going down the right path to correct this. In trying fix this problem I was clicking (in paperspace) on the Update Viewport Boundary icon (which, as far as I can tell, wasn't working) and then right mouse clicking and selecting Properties. That's the screen shot I posted. This time I first clicked on the selection tool, selected the viewport (paperspace) and right mouse clicked and them selected properties. This time the visible box showed up selected and unselecting it removed the outline!
Now sure exactly what the difference is between the two methods - I'll try to go back and figure that out - but... problem solved!