Why are "square law" devices important?
It's a particular classification of nonlinear behavior that has important applications. 

In the same way as you can consider, say, a BJT as linear over a limited range, you can consider something like a diode as square law over a limited range. That simplification allows you to analyze functions such as RF detectors analytically. 

See, for example, this Agilent paper "Square Law
and Linear Detection".