Hooke's law doesn't deal with the potential energy stored
in the spring. It only says that the distance you compress
or stretch a spring is proportional to the force applied to it.
The energy in the spring is the next step. It's the work you
put into stretching it or compressing it. Here ... look at this.
I think I can actually show it to you:
Work = (force) · (spring distance)
But Force = [ k · (spring distance) ] (Hooke's law)
So Work = [ k · (spring distance) ] · (spring distance) .
And there, Miss Moon, is where the x² comes from.
This is the most productive thing I've done
all day. I am so proud of myself.