People once thought that trying to square a circle was an illness What was the name of the illness?

The name is "morbus cyclometricus", coined by Augustus De Morgan, 1806-1871.

"Sanitas cyclometricus" becomes the contrasting coinage when considering the probable ancient "illness" of trying to prove that the earth was not flat. Or consider this example which promotes reasoning and imagination:

Inscribe a square within a circle so that two opposite sides are horizontal. Draw a straight line connecting the lower right corner with the upper left corner. Imagine the clockwise rotation* of the leftmost right triangle, keeping the left side length constant, the right side at 135 degrees and the horizontal side at 180 degrees.

The length of the horizontal side always defines the side of a square. And when a certain scalene triangle is created, this side defines the side of the square of the circle, visually proving that only 3 points on a circle are required to draw its square. This geometry may also provide proof that transcendental Pi can be represented by a geometric object!

* The triangle does not actually "rotate": the angle of the left side changes from 90 degrees to 45 degrees (close to 72.597 when the scalene triangle is created). To evaluate the scalene triangle, let the circle's diameter = 2000000 units; length of left side = square root of 2 x 1000000 and length of horizontal side = square root of Pi x 1000000.