The definitive answer on the absence of the infantry square has to do with the relative lethality of Civil War artillery vs Napoleonic. It would have been suicidal to form a square within range of a battery of 12 pound Napoleons.
The reason you don't read of classic Marshall Nye at Waterloo type attacks is the increase in range for both infantry & artillery during the Civil War. A line of horse walked to the 100 yard range of smoothbore muskets. They then went into a gallop & within 30 yards; changed to run in order to hit the opposing force in good order & achieve maximum shock. Attempting that within the 300 yard effective range of rifle toting infantry would have been suicidal.
Horses balk at stepping over bodies as well as smashing into a formed line sprouting bayonets. As a result, European war horses were trained to kick out & leap with viciously flinging hoofs. Needless to say, Civil War cavalry had no time for such old school frivolities.
The vast majority of Civil War cavalry were trained as dragoons. In effect, they were mounted infantry. Buford's command that opened the Battle of Gettysburg was trained to fight dismounted. Wilder's Lightening Brigade took the mounted infantry concept to its logical conclusion. They did not carry sabers; instead they had axes. They even picked the yellow cavalry piping out of the uniforms they were issued.
General Stanley, who was brought in to organize the AoC cavalry was a traditionalist. He insisted on long hours of saber drill. Just how many horses had their ears lopped off, a common saber training occurrence, is not recorded.