If the Origamido sheet is 16" x 20", then it's surface area is 320 inch². A 4" square has a surface area of 4"² = 16". Thus, from said Origamido sheet one can derive 320"/16" = 20 squares of 4". What Jared said was correct.
If the squares are 3" in length and width, they have a surface area that is 3"² = 9". Thus, 320" of surface area could provide you with 320"/9" = 35 5/9 squares. What Jared said was again correct.
If the squares have a surface area of 40 cm² - I assume that's what you mean - then you could derive ((16* 2,54)*(20*2,54))/40 = 51,6128 squares. Each square would have a length of 40^½, which is somewhere around 2,49". What Hydrax said was correct, though it's a bit of a peculiar choice.
Given the fact that you need 21 squares, the most optimal way to cut the Origamido sheet into pieces would be:
320"/x² = 21
21x² = 320"
x² = (320")/21
x = ((320")/21)^½ ≈ 3,9" in length and width.
3,9" is about 9,91 centimeters, thus it would satisfy Jared's condition that:
(...) Yeah, it's a little complex, but I would go for 7.5 to 10 cm squares.
All hail Origamido!