Multiplication and convolution
Here is your problem: "We know that squaring a signal doubles its bandwidth."

That is not true in general.
It is true for sinusoidal signals.

Probably you assume it must be true also for linear combinations of sinusoidal signal (i.e. Fourier sums/integerals which can be used to approximate a rectangular signal), but this is not the case. This reasoning works only for linear operations (e.g. multiplying by a constant, taking the derivative, integrating) but squaring is not a linear operation. 

(in other words: the linear combination of squares is not the same as the square of linear combinations)

Therefore the fact that your statement is true for single (pure) sinusoidal signals does not mean that it is also true for linear combination of sinusoidal signals (Fourier sum/integeral; e.g. rectangular signals).