How can I construct a square using a compass and straight edge in only 8 moves?
Your solution fails because you give up a move creating the center point of your bounding circle. To complete the challenge in eight moves, you need to use one of the preexisting intersections as your center point, and then rely on the fact that extending a line is not considered a move to create the fourth vertex of the square.
The following diagram illustrates these steps, where step 4b is the non-move extension of the line:
