The short answer is "No". Why? Because applying high gain to a signal not only forces the fundamental to run out of headroom (which would lead us to expect "squaring"), but also applies the same amount of gain to whatever harmonics/partials accompany that fundamental. Those harmonics do not have the same amplitude as the fundamental, so amplifying them by 100x or 200x or even 500x still leaves them below the clipping threshold. The result is that they are added to the fundamental, rather than displaced by it.
Stated bluntly, you're never going to get something out of a guitar string pushed to the limits that will sound like the square-wave output from an oscillator of the very same note. That doesn't mean it's "bad", just not identical. You will, however, get closer to an oscillator tone if the guitar is seriously filtered before applying gain, such that all those "contaminating" harmonics are removed. But, as you can imagine identifying the right filtering for every single note on every single string is not exactly stick-it-in-a-1590B stuff.