I started with the <strike>Tillehoff</strike> Lillehoff Sphere.
Can’t find the forum thread which discusses that, probably because I’m misspelling the man’s name, but it seems to work better for me with this one than the standard Sphere3D. His script had three settings, and I went with the “very high density” one.
(let’s see how much terminology I butcher by trying to cite this from memory…)
Edit mode.
Mirror along Y axis.
(it will feel like X in a moment)
Radial Symmetry on, with a value of 7.
Sculpt an eye socket. You now have seven evenly spaced eye sockets.
Select a pair of eye sockets, and build a face around that, watching for overlap from the surrounding faces.
To keep the animated GIF small in file size, I only rotate about 51 degrees (360 / 7), and the loop creates an illusion of continuous motion where you’re really just seeing one segment over and over again.
To keep that possible, I had to drop the idea of breaking things up with different facial expressions. But I will eventually rig this character up for multi-facial animation. 
Also, this is not a project for TextureMaster, though I guess you could clone from a background image or something. Hmm! I’ll have to play with this idea - it may prove more flexible than the vertex paint I was planning to use.
From here, I exported to 3D Studio, placed one eyeball, and used math to position and orient the other six that same distance from a central point.
And then, for some reason I thought rotating the actual model would be bad, so I set up a camera and some lights, and linked all of those to an invisible rotating thing. This may have been unnecessary, but I guess it gives me more flexibility with background objects later. That must have been what I was thinking.
Nothing too exciting going on with the materials at this point…
Again, no reason this couldn’t be done exclusively in ZBrush, though it may get pretty crazy with the multimarkers.