Why can we hear the full range of square waves from 1 Hz-20 kHz, but not sine waves of the same frequencies?
Most people can't hear sinewaves below about 20 Hz. The fact that you couldn't hear them until they got to 100 Hz is probably due to poor low-frequency response in your amplifier/speaker.

When you use a square waveform, you are adding a rich series of harmonics to the fundamental frequency &mdash; these are what you're hearing, and your brain extrapolates the fundamental frequency for you. Down to a point, that is &mdash; at some point you just start hearing individual clicks.