The circular aperture eliminates light from the edges, produces the most dramatic shift.
A shaped aperture accepts light from closer to the center and closer to the edges, mixes light from the different focal-length regions, hence spreads out the DOF.
OK, but a shaped aperture would have to be extremely "out of round" to have a noticeable effect. In most non-circular apertures, the portion of light transmitted through the irregular outer area is very small in comparison to the central part - even the triangular aperture mentioned earlier - so mixing of light from different-focal-length-edges regions is hardly greater than a circular aperture. Also, most non-circular apertures close down from all directions - light is eliminated from the edges - when stopped down so you are still going to get focus shift. I'd say on most commonly used lenses, the shape of the aperture has a negligible effect on focus shift.
If you are really going to minimize focus shift on stopping down you need an aperture which continues to transmit light across the entire radius on stopping down. For example a cat's eye which is near circular (wide open) for night vision but closes down to a slit in bright light. Such an aperture would produce unusual DOF effects, since the DOF in the vertical direction hardly changes on stopping down, but increases greatly horizontally (hmmm... could be interesting).
Another example are the "sink strainer" stoppers used with some large format cameras. In some ways these act a bit like a neutral density filter in that they reduce the amount of light passing but still transmit across much of the aperture radius, so the DOF is not altered greatly.
