Nah. It's not like that theorem implies that we must square errors, based only on axioms which are facts of the natural world.
On the contrary, squared errors are part of the formulas taken as axioms from which to derive the theorem.
I wouldn't be surprised if we could derive a theorem whose essence is tantamount to the essence of the famous Central Limit Theorem, using some other arbitrary formula. Like error to the fourth power.
My point was that many things are normally distributed (likely due to the CLT) and the square error is the right choice in those cases. This is an empirical thing as well, in that measurement noise is often Gaussian.
On the contrary, squared errors are part of the formulas taken as axioms from which to derive the theorem.
I wouldn't be surprised if we could derive a theorem whose essence is tantamount to the essence of the famous Central Limit Theorem, using some other arbitrary formula. Like error to the fourth power.