The first strategy assumes you want to maximize your chance of winning something for each game you play. You can do this by picking squares along a diagonal or otherwise such that each square selected has a unique row and column. You minimize the effect of the correlations between the squares in each of the quarters and so you minimize the chances of winning two or more squares. You can maximize this strategy on a per square basis by picking a single square.
The second strategy assumes you want to maximize your chance for winning two or more squares. The strategy essentially trades off a small increased probability that you won't win anything for a much larger (relative) probability that you'll win two or more squares. You can adopt this strategy by picking squares in the row or column corresponding to the team that is the underdog.
To highlight the two strategies, one study showed the following: Picking seven squares in a column gives you nearly the same chance to win something as picking six along the diagonal, but a 75% more chance of winning 2 or more squares.
Here is a grand tabulation of all the possible ways of selecting 1, 2, 3, or 4 squares ordered by the strategy probability for winning 2 or more squares. This tabulation is based upon 1 billion simulations using 5209 games (1994-through the the 2013-2014 regular season). All shown are all the possible ways of selecting 5 squares. Squares in adjacent columns do not need to be next to each other, but anywhere along the same column where the column represents the underdog. Similarly for the rows. If you are picking four squares, you can choose the first strategy (four in a diagonal, labeled A) to give you a maximum probability of 13.47% to win something or the second strategy (four in a column, labeled H) to give you the maximum probability of 2.97% to win two or more squares. Or ... you can choose some other configuration to give you the best of both worlds. The choice of strategy is up to you!