First, it is important to understand exactly what is being claimed. It is impossible to start with a circle and produce a square of the same area using only straight-edge and compass constructions starting with the circle.
The reason it is impossible is based on the fact that each time you find the intersection of two lines, you are essentially solving a linear equation; each time you find the intersection of two circles or a line and a circle, you are solving a quadratic equation. So, a ruler and compass construction corresponds to a sequence of solutions of linear and quadratic equations.
Well, it turns out that the ration pi, that of the circumference to the diameter of a circle cannot be exactly found by this type of construction. The proof of THIS is quite difficult and was solved by Hermite in the late 1800's. But, if you were able to find the square, you would be essentially finding a line of length the square root of pi. If you could do this, finding one of length pi would not be difficult.