If you knew which rows and columns would get assigned which digits, then you'd want the best numbers for each of the quarters (Q1-Q4) and the final score (FS). Here are the probabilities that have shown up using 5220 professional football game box scores (quarter by quarter scores) from 1994 (when the two point conversion rule was adopted) through the entire 2013-2014 season.
This shows that overall for Q1-Q3,FS, the numbers 0 (especially in the first quarter) and 7 are the best, 3 and 4 are good, 6 and 1 are OK, and 8, 9, 2, and 5 are not so good (unless the score that day happens to turn out to be 42 to 35!).
But the overall numbers only tell part of the story. You have a combination of numbers - one number for each team. Depending on which of your numbers corresponds to the favorite and underdog team does matter. Here is an expanded table showing the relative value of all the possible combinations of digits for Q1-Q3,FS. This table shows what your numbers are worth relative to the others. The 0-0 and 7-0 (favorite-underdog) squares are the best and worth about 8x the average square. The 2-2 rarely shows up and is worth only about 1/40th (0.02) of an average square's value.
The bottom line is that Football Squares a fair game, such that over the long term, it makes no difference as to what squares you select in terms of how many winning squares you are likely to collect. Someone who has no knowledge of football is expected to win just as many squares as anyone else choosing the same number of squares. This bottom line, however, does not mean that you can't use a strategy. A strategy can be used to affect the likelihood of certain outcomes in any one game. In short, you can trade a small increased chance that you won't win any squares for a much larger chance that you'll win two, three, or even (although rare) four squares in a single game.
Using the 5220 professional football game box scores (1994-2013 regular season), over one billion computer simulations have been run where squares are picked, a game is chosen, and rows and columns assigned random 0-9 digits. It turns out there are 60 unique configurations you can select up to five squares given the digits for rows and columns are random. Over a billion simulations are needed because there are 4,124,276,932,608,000,000 total possibilities (10! x 10! x 5220 x 60 - that's over 4 quintillion or 4 billion billion).
At the extreme ends, there are two strategies that emerge. If you pick squares such that each square selected has a unique row and column, you will maximize your chance at winning something (most often only a single square) and minimize your chance, relative to other strategies, that you'll win two or more squares in a single game.
The other extreme is to select a strategy that maximizes your chance of winning two or more squares. You can try to go home the BIG winner (I like this strategy!). This strategy says that you should select squares along the row/column corresponding to the visiting team (or with even stronger results, use the row/column corresponding to the underdog especially if the game is expected to be low scoring). In fact, you should pick as many squares as you can in that row/column and even start on a next row/column as long as you aren't violating any etiquette for picking too many squares.
Consider selecting four squares in the two extreme strategies ... configuration H, picking them all in a column - maximizes winning two or more squares and configuration A, picking them with unique rows and columns - maximizes winning something. Expanded, the probability table looks like this:
The expected squares are calculated by multiplying the number of wins with the associated probability and summing it all up. The numbers for 2 or more wins and winning something (>0) match the values next to the patterns below. All configurations of selecting four squares sum to 16% as each square has a 1% expectation of winning each of the Q1-3,FS (4 squares x 4 opportunities x 1% each is 16%). Notice, that essentially your strategy - how you select your squares - allows you to trade off a small probability for winning something for a much larger probability for winning two or more squares.
All possible ways to select up to 5 squares
Below is a final tabulation of all possible ways to select 1-5 squares along with the probability of winning 2 or more squares and the probability of winning something. Reading top to bottom and then left to right, the configurations are ordered to maximize your probability of winning two or more squares. The column direction reflects the underdog team and the row direction reflects the favorite. Squares next to each other mean any squares on the same row or column. A square offset from the others mean any square with a unique row and column. Pick one of the extreme configurations or pick one in between - the choice of strategy is yours
The Bottom Line
In summary, your strategy can't change your total expected value only the probability for various outcomes to happen. However, your strategy can definitely change the likelihood for the different outcomes. For me, I want to go home the big winner, so I'll live and die each time I play by selecting them all in a column corresponding to the underdog team. More times, I'll go away without winning, but when I do win, I'll be the big winner of the day! The choice of strategy is yours!