Very similar to the proof of Sperner's Lemma, just being a rectangle instead of a triangle. Consider each unit square a room and each edge between two rooms a wall. An outside area is also considered a very big room. A wall have a door on it iff two vertices of that wall have colors 1 and 2. We can see that a (small) room has four vertices in all different colors if and only if it has exactly one door. Moreover, a (small) room cannot have exactly three doors. Therefore, a room has four vertices in all different colors if and only if it has odd number of doors. Consider the outside very big room. Doors can only exist on $AB$. Morever, $A$ is color 1 and $B$ is color 2, so by parity there are odd number of doors on $AB$. Consider a graph with each room as a vertex, and two rooms have edge iff there is a door between them. The vertex of the outside very big room has odd degree. Hence, except for this vertex, there are odd number of vertices with odd degree. Therefore, there are odd number of rooms with four vertices in all different colors. In particular, there is at least one of such room.