There was an invisible ship bug in TA a while back-definitely pre-1.96, maybe pre-1.95, which I believe was fixed.
If you're seeing this in all GCII versions, Scout_Pilot, my guess is you have either old drivers or a graphics option turned on that your video card can't handle.
Suggestions for both of you follow.
A few things to check:
-uncheck enable animated components (only exists in TA)
-uncheck use enhanced ship rendering (DA/TA)
-uncheck allow mixed vertex processing mode (DA/TA)
-uncheck any or all options beginning with Disable under Performance Options in Video
Doing the last thing on the list will make the game look worse, but doing so got it to run on my dad's system, which only has SiS integrated 740M graphics-which should be too ancient to run it.
Failing all of the above, start a new thread with your issue, Scout_Pilot, and each of you post your debug.err in your respective threads. It is located in the following places, based on expansion (each expansion creates a separate debug.err).
\Program Files\Stardock\TotalGaming\GalCiv2 or
\My Documents\My Games\GC2DarkAvatar or
\My Documents\My Games\GC2TwilightArnor
I would assume Endless Universe has DA and TA partitioned the same way, but I don't know for certain.