For any one football game with four winning squares after Q1-Q3, and after the final score (FS), either you win zero squares, 1 square, 2 squares, 3 squares, or 4 squares. By making choices on the configuration of which squares you choose, how many squares you choose, and even whether you play certain games or not, you can adjust how likely each of those categories turn up. In essence, you can trade a small increase in the chance that you win zero squares for a greater chance that you win 2, 3, or 4 squares (2 or more).
If your strategy is to play the game and minimize the chance that you win no squares, then you should pick your squares on unique rows and columns. With this approach, you'll also have the highest probability of winning exactly one square. The downside, is that you'll have the lowest probability of winning 2, 3, or 4 squares. There is merit to this strategy.
Another strategy that I will focus upon is one where you maximize your chances of winning big by taking 2, 3, or all 4 squares that day even if means increasing the chance that you walk away with no winning squares. You have the best chance to claim total victory over all the other players ... I like this strategy.
The first simulations show a clear winning way to play your squares for this strategy ... pick them all in a column corresponding to the visiting team. Picking them all in a row corresponding to the home team isn't too bad either, but especially to maximize the chances for the rare day of winning all four squares, pick them all in a column.
Graphically, and using numbers from multiple simulations, one billion trials, here are the results for picking four squares:
The chart has a horizontal axis corresponding to the order of the players as shown for the different configurations based upon the likelihood of winning two or more squares. The all vertical choice (Player H) listed first and the choice of picking squares in unique rows and columns (Player A) listed last. The blue line represents the probability of winning at least one square with that choice of configuration. This is smallest for Player H at 12.46% and rises to 13.46% for Player A. The darker blue line, representing the strategy to try to win two or more squares is shown to be maximum for Player H at 2.92% while Player A's chances for this are 2.24%. Even more dramatic is the green line, that utilizes the zoomed in right axis, showing the (small) chance for claiming total victory and winning all four squares ... Player H at 0.062% and Player A at 0.014%. So Player H trades a little bit of the blue line (winning at least one square) for a higher probability of winning two or more squares.
Comparing the strategy to be dominant and have the best chance to win at least two squares, Player H to Player A, Player H reduces the chance of winning anything at all by only about 7.4% but then increases the chances of winning two or more squares by 30% along with a 337% increase (more than 3x likely) for the rare chance that all four squares are to be won!