jmc

The prime factorization of $1200$ is $1200=2^4\cdot 3\cdot 5^2$. Thus, the factors of $1200$ that can be written in the form $n^2$ are $1^2$, $2^2$, $4^2$, $5^2$, $10^2$ and $20^2$. The sum of these values of $n$ is $1+2+4+5+10+20=\boxed{42}$.