Method 1:
Given one side, rotate it 90 degrees at both A and B, construct endpoints C and D, connect the endpoints. To test the construction, try to move the square at A or B, the entire square will change to maintain the equal lengths and right angles.
Method 2:
Given one side, AB, construct perpendicular lines to AB at both A and B. Using an arc, measure the length AB, and mark off one of the perpendicular lines at that length.
This point can be labeled C, from C we can construct either a parallel line to AB or a perpendicular to the constructed line AC. The construction is then cleaned up by constructing segments AC, CE, and BE, and hiding the lines used in construction.
Other methods can vary the order in which the segments are constructed, and whether they are perpendicular or parallel to specific segments, either given or constructed.
For example, at A construct a perpendicular to AB, measure AB on that line, label that point C. At C, the intersection, construct a parallel to AB, measure AB on that line, AC, and label that point D. At D construct a parallel to AC or a perpendicular to AD.