Several things spring to mind. I’m ordering these in terms of increasing math level: the last two only make sense if you’ve taken calculus, which most five-year-olds haven’t.

1) Areas are measured in length squared. Quantities that involve area coverage, or “stuff” spreading out over a certain amount of space, tend to involve squares or inverse squares. (Examples: gravity, electricity, fluid flow)

2) The square of a quantity is symmetric about zero: a negative quantity squared is positive. So if a phenomenon depends on how *much* something changes, but not which *direction* it changes in, a square will often appear. (Examples: energy)

3) We often successively approximate complicated functions using simpler ones via Taylor series: we assume the function is constant, plus a linear slope, plus a squared term … and often that’s accurate enough and we don’t need to go to cubes and higher. And because of point #2, the linear slope will often be zero. (Examples: pendulums, wave speeds)

4) Second derivatives seem to hold a special place in the universe — Newton’s Law, Maxwell’s Equation, the Schroedinger equation, all involve a second derivative. Often we make the simple approximation that some quantity (like, say, acceleration) is constant. This means that the quantity it’s a second derivative of involves a square.

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