Suppose we have sets $A,B,C$ which are $n$-equidistant.
Then $n=|A\triangle C|=|(A\triangle B)\triangle (B\triangle C)|$. But $A\triangle B$ and $B\triangle C$ are finite sets of size $n$, and the size of the symmetric difference of two finite sets of equal size is always even.
Why? If $|X|=|Y|=n$, then $|X\triangle Y|=|X\setminus (X\cap Y)|+|Y\setminus (X\cap Y)| = 2(n-|X\cap Y|)$.
Thus $M_n=2$ whenever $n$ is odd.