Tony's right. It's a cosmetic gimmic that has some following in the car-audio crowd (not surprising). You'll notice that none of the serious home hifi and home theater subs use square drivers. Conventional round cones are lighter, stiffer, and have fewer breakup modes. All else being equal, they sound better.
As gimmicky as it may be there is a reason for it in car audio since SPL competition began. Increased Sd (radiating surface), which translates into higher Vd (volumetric displacement). Instead of Pi*r^2 you have l^2. If 1/2 l is equal to r then the square will always have greater area with drivers the size of subwoofers.
How light the cone is will depend more upon the material used rather than it being square or circle. However, with greater surface area usually there will be greater weight.
Concerning stiffness, I've seen a few square subs take 150+ dB. In home audio I doubt cone stiffness will be a large concern.
Have you seen testing that shows break-up from square subs below 80hz? I have yet to see one of the kicker subs be peaking from break-up in it's operating region.
There is one Home Audio sub I know of, which uses a square cone (in this case it operates as the complete baffle also).
Adire had the motor mechanism sitting on their counter with a 9v battery next to it. Besides this thing being So ridiculous that I hardly recognized it as a sub driver's motor, it has like 4 VC i think, and using a 9 volt battery it moves inches. It's crazy. It's not a driver I look forward to hearing, because that thing would be completely nuts. Is my point. That things is completely insane.