Well, a square wave is really just a bunch of sines on top of each other.
Almost correct. A square wave is actually a sine wave of the same frequency but with all of the ODD-harmonics of that frequency added in. The magnitude of each odd harmonic decreases as the frequency rises. Adding the odd-harmonics "files In" the corners of the sine wave to make it a square wave.
Long story short, a 60 Hz square wave will contain a 60 Hz sine wave, a 180 Hz sine wave, a 300Hz sine wave, etc, etc etc. Transformers tend to work only a specific frequency so the they won't react well the 180Hz and the other high frequencies. The harmonics will be mostly filtered out and generate heating in the transformer. The transformer output will be 60 Hz with few if any of the harmonics so it will be a nearly pure sine wave.
Note: transformers can be built to have wide frequency range, for example for audio use, but power transformers are usually designed for a specific frequency and will sharply attenuate other frequencies.