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!
