The squares are push buttons for Apollo's giant green hand thing. Pressing one makes the ship play a sound clip.
That's an interesting theory, Greg, but I remember Apollo's fingers to be rather big and the squares apparently don't stick out far enough to be pushed.
Considering that many Star Trek designers had very fond memories of Nemo's Nautilus
from "20,000 Leagues Under the Sea" (including Andrew Probert and Greg Jein), we might rather be looking at an electro-shock mechanism that prevents interstellar life forms from attaching themselves to the upper side of the saucer - and blocking the main screen view.