Actually, perfect squares are extremely rare in nature. Squares are human constructs.
We imagine Pythagoras and the Greek philosophers drawing their squares in the sand, dividing the sides equally, connecting the divisions, and counting the resulting smaller squares to develop the concept of a number multiplied by itself. The concept is probably much older. Both the Babylonians and the Egyptians had methods for calculating volume based on squared empirical approximations of Pi.
It is easy to imagine our forebears drawing a square around a circle to get the notion that the area of the circle varied with the radius squared. If that were the end of it one could dismiss it as the artifact of a practical approach, but these squares keep coming back.
Newton’s laws of gravitation vary the square of distance. Gravity? We’re a long way from geometric lines in the sand now.
Yagahhdabekiddnme! Why not the cube root or something…