I believe the answer is "no." Count the vertices (corners) that have an odd number of line segments meeting. If there are more than two such corners, and the total number is an even number, then the drawing cannot be traced without re-using a line. In this case, there are four vertices with 5 lines each (the corners of the square).
Yes , you can draw this without overdrawing it. I used to do this in my school days.. you need to work (draw) through this using a paper and folding it in half to come to the other side were you left. I hope this may help..it's just lil tricky. DIY!