Suppose such a labeling exists. Let $F_k$ the region formed by the cells labeled $\{1,2,\dots,k\}$. There is a minimum $k>1$ such that $F_k$ connects two opposite sides of the checkerboard (wlog suppose left and right side). Color in black the cells belonging to $F_{k-1}$, and in white the rest.

Then, $k$ is in a white cell touching a black cell, and in every other column there is at least one black and one white cell. So overall there are at least $n$ white cells touching black cells. One of them must contain a number $\geq k-1+n$, and (since it touches the black region) it touches a number $\leq k-1$. Contradiction!

Suppose such a labeling exists. Let $F_k$ the region formed by the cells labeled $\{1,2,\dots,k\}$. There is a minimum $k>1$ such that $F_k$ connects two opposite sides of the checkerboard (wlog suppose left and right side). Color in black the cells belonging to $F_{k-1}$, and in white the rest.

Then, $k$ is in a white cell touching a black cell, and in every other column there is at least one black and one white cell. So overall there are at least $n$ white cells touching black cells (one per column). One of them must contain a number $\geq k-1+n$, and (since it touches the black region) it touches a number $\leq k-1$. Contradiction!

Suppose such a labeling exists. Then there is a minimum $k$ such that the region formed by the cells labeled $1,2,\dots,k$ touches two opposite borders of the checkerboard. Without loss of generality, we may assume that $k$ touches the topmost border of the checkerboard (otherwise, rotate the checkerboard).

Let us color in black the cells labeled $1,2,\dots,k-1$. Then, in every row there is at least one white cell adjacent to a black cell, so at least $n$ overall. At least one of these cells must contain a number $\geq k-1+n$, and (since it is adjacent to the black region) it is adjacent to a number $\leq k-1$. Contradiction!

Suppose such a labeling exists. Let $F_k$ the region formed by the cells labeled $\{1,2,\dots,k\}$. There is a minimum $k>1$ such that $F_k$ connects two opposite sides of the checkerboard (wlog suppose left and right side). Color in black the cells belonging to $F_{k-1}$, and in white the rest. 

Then, $k$ is in a white cell touching a black cell, and in every other column there is at least one black and one white cell. So overall there are at least $n$ white cells touching black cells. One of them must contain a number $\geq k-1+n$, and (since it touches the black region) it touches a number $\leq k-1$. Contradiction!