It would have been better if they looked for the densest circle with an area of 1 square km. Although a square shaped area has its tangible benefits - you "feel" population density more when it’s on your own street obviously, and since square areas provide more information along a few axes (the ones containing the corners), you can get an idea of the population density in an area by placing a square’s corners on the street. A square is probably not the best shape for this, though.

You certainly would.

Just think how insanely unlikely it would be for this not to be the case.

One could also increase the density by allowing non square shapes or squares not fixed on the grid and so on. A more general approach would always lead to slightly higher scores.