Gain of Integrator Amplifier With Square Wave Input
I would think that since a square wave is composed of a sine wave at
  the primary frequency and a number of odd order harmonic frequencies,
  there must be a way to add to this and solve it more accurately.


A square wave comprises odd order harmonics that have amplitudes (relative to the fundamental) of 1/3, 1/5, 1/7, 1/9 etc..

An integrator attenuates higher order harmonics more than lower order ones. This attenuation is linear with frequency i.e. assuming the fundamental is the reference point, the 3rd order harmonic will be attenuated by a factor of 3:1. The fifth order harmonic is attenuated by 5:1.

So now, the relative values of harmonics to the fundamental are: -

1, 1/9, 1/35, 1/49, 1/81 etc..

This of course is a triangle wave: -



So, you input a square wave into an integrator and you get a triangle wave out. How you relate this to "gain" is up to you.