two different sides,
one corner and an opposing side, or
two opposite corners of the square.

Sequential slices may or may not cross previous ones, but a set of slices will subdivide the square into polygonal regions.

Find (or prove impossible) a way to slice a square into 7 pieces of equal area with n distinct slices for each n={3,4,5,6,7}

A concrete construction of the n=4 case:

Start with square ABCD.  Add point E on AB so that AE=3/7; points F, G, and H on BC so that BF=1/2 and GH=HC=17/84; point I on CD so that ID=3/7; and points J and K so that KJ=JD=31/84.  Draw lines EF, GK, HJ, and EI to create the seven areas.