Geraldine and Suzanne sent in their solutions to this problem. They said:

We think there are five different squares which you can make on a 4 by 4 geoboard. Here is a picture of three of them:



Here is a picture of the other two which only have pegs at their corners, not along their sides:



The pink square in the second picture has an area of two square units. We don't think you can make a square with an area of three square units as the next size up square we have has sides of 2 which means its area is 4 square units (too big).