An Ebborian named Ev'Hu suggests, "Well, you could have a rule that world-sides whose thickness tends toward zero, must have a degree of reality that also tends to zero. And then the rule which says that you square the thickness of a world-side, would let the probability tend toward zero as the world-thickness tended toward zero. QED."

An argument somewhat like this except not stupid is now known. Namely, the squaring rule can be motivated by a frequentist argument that successfully distinguishes it from a cubing rule or whatever. See for example this lecture. The idea is to start with the postulate that being in an exact eigenstate of an observable means a measurement of that observable should yield the corresponding outcome with certainty. From this the Born rule can be seen as a consequence. Specifically, suppose you have a state like a|a> + b|b>, where = 0. Then, you want to know the statistics for a measurement in the |a>,|b> basis. For n copies of this state, you can make a frequency operator so that the eigenvalue m/n corresponds to getting outcome |a> m times out of n. In the limit where you have infinitely many copies of the state a|a> + b|b>, you obtain an eigenstate of this operator with eigenvalue m/n = |a|^2.