Why is power of a signal equal to square of that signal?
If the signal is represented as a voltage \$v(t)\$ or a current \$i(t)\$ and it is connected to a (1 ohm) resistor, the power dissipated in the resistor is proportional to \$v^2(t)\$ or \$i^2(t)\$.
Apart from that, defining power as a positive, increasing function of the signal amplitude has useful mathematical properties.