First, I'll assume you're stuck with square, otherwise you really need to consider making it round so it can't be dropped in the hole.
For maximum moments and shears, check out Roark's Formulas for Stress and Strain. They have that case in there (uniform load, pinned on 4 sides).
I think SperlingPE stated it best, however. Find a Neenah catalogue and buy one. It will probably be lighter than the one you come up with.