While I still can't answer in the general case, in the case where n > 2 and Alan moves only with 0 in attempt to fill a row or column with 0s, he cannot win.
This proof is written semi-informally for ease of reading.
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Call Alan's moves 0s and Barbara's moves Xs.
A "blocked" row or column is a row or column that Barbara has at least one X. A "unblocked" row or column is free of Xs.
For Alan to win on an nxn grid, after his move is complete there needs to be at least one row with n-1 unblocked 0s and at least one row with n-1 unblocked 0s.
Define a set R which contains, for each unblocked row, the number of 0s on that row.
Define a set C which contains, for each unblocked column, the number of 0s on that column.
For our the explanation that follows we will write set R followed by set C. For example, if R={2,1} and C={1,3} the sets will be written as {2,1} {1,3}.
The game begins with both R and C as the empty set.
(1st move) Alan moves and the sets become {1} {1}.
(2nd move) Barbara moves to block a row and the sets become {} {1}.
(3rd move) Alan has three choices, let us consider each:
[Case 1] Alan moves in the same column as the existing unblocked 0. The sets become {1} {2}
Barbara moves in the same column as the two unblocked 0s. The sets become {1} {}.
[Case 2] Alan moves in a square that is blocked in both row or column. Barbara blocks the unblocked column and the sets become {} {}.
[Case 3] Alan moves in a square that contains no 0s or Xs in both the row and column. The sets become {1} {1,1}.
Now the situation is as in the diagram above. Suppose the 0s are in A and D, and B and D are empty. One of the rows must be blocked; suppose it is the same row as A. Then Barbara moves at C and the sets become {0} } {1}. If the blocked row is the same row as D, Barbara moves at B and the sets become {0} } {1}.
The situation if the 0s are at B and C is symmetrical.
Now note that all the cases are either identical to an earlier position of the game or are symmetrical to an earlier position. Therefore Alan can never win the game.
