A post to answer my own question, in case anyone finds this topic in a search.
I have installed a test Catalina system (10.15.7) and upon re-licensing my version of Eagle 7.7.0 ultimate, I find that it works just fine. I have read somewhere else that a user found Eagle to crash if he zoomed too quickly, but I noticed no such problem.
I opened some libraries and some of my layouts, and everything seemed to work just fine. I am not going to keep Catalina, since all 32 bit stuff goes away, but Eagle 7.7.0 is a 64 bit app, and it seems to work just fine under Catalina.
I will be keeping Mac OS Mojave (10.14.6), and Eagle 7.7.0 works just fine there too.