I often see photographers while they're about to shoot, they move their cameras in a rapid square motion while having the subject in their camera before proceeding with the shot.
What am I missing? Is it something to do with the Rule of Thirds?
There is a concept called "focus and recompose" which is typically used with cameras that have a limited spatial distribution of auto focus (AF) points or only one really good auto focus point in the center of the frame.
If this is the case and the photographer wants to photograph a subject that is not incidentally at the same location within the frame as the auto focus point he can do the following procedure:
Note: This technique doesn't really require what one would consider a square motion. The movement may be executed in a pure straight line. But it may look different.
There is also a chance that the photographer simply doesn't want to change AF points between shots or that he isn't able to because it would last too long to fiddle with the controls.
It could indeed be reasoned in the photographer wanting to fulfil the rule of thirds. But it could also apply in other situations. Many SLR cameras have a rather bad distribution of AF points. (which is trending to get better in future)
A second reason for this behavior could be that he wants to use exposure spot metering which often is only possible in the center of the frame. The same principles apply here as above.
Additional Info: If the photographer uses a SLR in LiveView mode or a mirror-less camera the chances are high that he does not have to fall back to focus&recompose because AF points in this setting are distributed equally over almost the whole sensor area.
Many cameras don't have a full coverage viewfinder. That is, the viewfinder doesn't show the complete frame that will be captured by the image. By moving the camera to each of the four sides of the frame the photographer can check the edges to be sure there is nothing there they don't want in the picture.
It could be that you are observing the photographer using the Brenizer Method (see also Ryan Brenizer's site) in action. Essentially, the Brenizer Method uses panoramic stitching to create very shallow depth of field effects for portrait photography. The photographer takes overlapping photos in a rectangular fashion, often more than just 4 or 5, and combines them with stitching software.