CM is a linear calculation. Gravity is a inverse square.
Think about a small test mass at the center of your square.
All the forces are near balanced, not much going on.
Replace the corners with 4M at the center, and the test mass dives in.
Va
A -->
..
Vb^ . .
| . .
|. .
B D
\ /|
\ / |
\ / v Vd
\/
<-- C
Vc
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