Paul McKellar is one of Square’s founders. In a blog post from January 2011, he gave 3 reasons to explain his decision to join Jack Dorsey at the beginning of Square. They are well worth reading, but this bit about the card reader was unknown to me:
Square’s card reader uses the actual motion of the card swipe to generate a sound, which is decoded through a microphone, and processed in real time using the phones internet connection.
It is so far from what I had imagined. That explains why you plug the reader in the headphone jack.