If $n=k$ then the only case pigeon wins is obviously $m\geq n^2$ since after seagull's turns all the eggs in the grid are eaten. Now let $n\geq k+1$ I'll show that for every $m\geq k^2+1$ pigeon wins and for $m\leq k^2-1$ seagull wins. Firstly assume that $m\leq k^2-1$. Then seagull can pick a fixed $k\times k$ rectangle and each time eat all the eggs inside them. In that way pigeon can't wins because $m<n^2$ ( so it can't wins only in one move) and then in order to complete some time the $k\times k$ gap that seagull makes has to lay at least $k^2$ eggs which is impossible since $m\leq k^2-1$. So if pigeon wants to win has to lay each time $m\geq k^2$ eggs. If $m\geq k^2+1$ let the birds play and pick the configuration where the number of eggs is maximum,say $t$. If $t=n^2$ pigeon wins. Otherwise seagul plays and eat the eggs in a $k\times k$ rectangle. If there is a grid not in this rectancle that not contains an egg then pigeon fisrtly can lay an egg in each grid inside the $k\times k$ rectangle and then lay an egg on it. But then we have one more egg that before, contradiction because of our choice of configuration. Hence all the grid outside the $k\times k$ rectangle contains at least an egg and the pigeon can lay an egg in each grid inside the $k\times k$ rectangle and wins. So for $m\geq k^2+1$ pigeon wins. The only case left is $m=k^2$. I'll show that pigeon wins if and only $n\leq 2k-1$. Suppose that for some $n$ pigeon wins, then for $n-1$ also wins (simple), so it's enough to show that pigeon wins for $n=2k-1$ and loses for $n=2k$. Take $n=2k$ and suppose that after a pigeon's turn there are $a\leq n^2-1$ grids that contain at least one egg. Then seagull can remove the eggs from at least $a/4$ that grids and then pigeon can lay an egg to at most $k^2$ more grids and so the new number of grids that contain at least one egg is $b=\leq a-a/4+k^2=3/4a+k^2\leq 3/4(n^2-1)+n^2/4<n^2$ so $b\leq n^2-1$ and inductively pigeon never can fills all the grids. Now assume that $n=2k-1$. Then there is a unique grid (in the center of the board) that belongs to all $k\times k$ rectangles. Pigeon colors this grid black and starts lay eggs in the grids but not in that black, after each seagull's at most $k^2-1$ grids empty and thus pigeon can fill some time all the grids except the black one. Then seagull play removing eggs from at most $k^2-1$ grids and pigeon lay one egg in each of them and also in the black and wins. Summary : if $k\leq n\leq 2k-1$ then pigeon wins for $m\geq k^2$ and if $n\geq 2k$ pigeon wins for $m\geq k^2+1$