Chase, the use of squares of infantry (hollow or otherwise) goes back to classical times, when the ancient Greeks used that formation to counter Persian cavalry and charioteers. Of course, in those days before the development of firearms, the weapon used to keep a mounted force away was an extremely long lance. Through extensive drill and discipline, the Greeks could raise their lances, shift fronts, and lower their lances in a new direction almost immediately. For the Persians, it was like trying to wrestle with a porcupine. No matter what direction they tried to attack from, they kept running into those long lances. Before the Greeks came up with this tactic, infantry deployed on open, level terrain was almost always at a disadvantage against cavalry.

The Romans took it a step further, and used their infantry to form a hollow square around companies of archers and slingers. This way they could reach out and touch someone before the enemy cavalry got close enough to engage, hopefully reducing their numbers so that when the cavalry did close on them, it would be more of an even fight. The archers, who generally wore no armor, would be safe within the porcupine, while the armored infantrymen forming the square did what they were trained to do.

The development of firearms pretty much made this tactic obsolete. It's last significant use was during the Napoleonic wars, when cavalry was still using swords and lances as their primary weapons. By the time of the Civil War, cavalry was equipped with firearms, often with repeating rifles, and used different tactics. It was sometimes used by Union regiments fighting the Indians on the plains, when the occasion called for it, but as a matter of military doctrine its day had passed.