To do and watch:

Place the square wheel on the far end of the road as marked.  Spin the wheel quickly.

Watch as the wheel rolls along smoothly.  The axle stays at a constant height.

Why does this work:

As the square wheel rolls down the road, the center of gravity is always at the same height.  The corner of the square will roll over the lowest point of the road while the sides’ midpoint will roll over the highest point of the road.  Therefore, the axis remains at the same height and rolls smoothly.