Another way to see this is by using Pontryagin-Thom.  The generator of $\pi_3(S^2)$ is represented by an unlink in $\mathbb{R}^3$ which twists around once (like a figure-8).  So precomposing $\eta$ with an element of $\pi_3(S^3)$ is a disjoint union of n figure-8's, while postcomposing with an element of $\pi_2(S^2)$ is *cabling* the figure-8 n-times.  The n-cable of the figure-8 has $n^2$ crossings, so it's cobordant to $n^2$ figure-8s.

The way I think about this is that the figure-8 is explaining a recipe for turning a 2-loop into a 3-loop (i.e. appear the loop and its inverse, braid them past each other using Eckman-Hilton, and then cancel).  If you first do the Hopf construction and then take its nth power as a 3-loop you're first doing a figure-8 and then taking its disjoint union n-times.  While if you take the power of a 2-loop and then apply the Hopf construction you're first taking the disjoint union of n points and then doing the figure-8 construction to all of them together.  This is why it's composing with an element of $\pi_2(S^2)$ which corresponds to the cabling.