The concept of a perfect square wave is one that is
simultaneous with the concept of infinite energy
in pretty much any domain.

You could impress a true square wave on a record
surface, by any number of means OTHER than with
a standard cutter.

But now let's look at the requirements that imposes
on the stylus: it must change position between the
minimum and maximum in zero time. That means
it's moving at infinite velocity. Since in Newtonian
mechanics, e = 1/2 mv^2, the kinetic energy imparted
on the stylus is infinite. In Einsteinian electrodynamics,
this situation is not even remotely possible, as the
fastest velocity relative to any observer is the speed of
light, and any finite mass traveling at the speed of
light possesses infinite energy.

And let's even pretend that's possible. The output
is the differential with respect to time of the stylus
position, which means the cartridge must put out
infinite current into that 47 kOhm phono preamp
and that dissipates infinite p=power since P = i^2 R.

And even if we dealt with that, assume the lowly wire
connecting the cartridge to the preamp had infinitesimal
but non-zero capacitance: propogating voltages whose
rate of changes are infinite requires infinite current,
since I = C dv/dt.

In any case, I would not concern myself much with
the square wave performance of LP systems: first,
true square waves are impossible and second, even
the imperfect ones are pretty useless.