A regular pentagon may be cut into as few as six pieces that will fit together without any turning over and form a square, as I shall show below. 
The best answer has been in seven pieces.

The pentagon is ABCDE. By the cut AC and the cut FM (F being the middle point between A and C, and M being the same distance from A as F) we get two pieces that may be placed in position at GHEA and form the parallelogram GHDC. 
We then find the mean proportional between the length HD and the height of the parallelogram. 
This distance we mark off from C at K, then draw CK, and from G drop the line GL, perpendicular to KC. 
The rest is easy and rather obvious. It will be seen that the six pieces will form either the pentagon or the square.