Solution:

Consider the 4 points shown in the diagram. In each case the segment joining them is the diagonal of an $m \times 1$ parallelogram or rectangle, so it cannot contain any other lattice points. The next points along each line are obviously outside set $D_n$. That proves the first part.

The second part is the standard parity argument. Color $(x, y)$ with color 1 if $x$ and $y$ are both even, 2 if $x$ is even and $y$ is odd, 3 if $x$ is odd and $y$ is even, and 4 if $x$ and $y$ are both odd. Then if two points are the same color, that means the first coordinates are the same parity and their second coordinates are the same parity. Hence the midpoint of the segment joining them is also a lattice point and they are not the only two points of $D_n$ on the line.