jmc

If $n^2+12n-2007=m^2$, we can complete the square on the left-hand side to get $n^2+12n+36=m^2+2043$ so $(n+6{)}^2=m^2+2043$. Subtracting $m^2$ and factoring the left-hand side, we get $(n+m+6)(n-m+6)=2043$. $2043=3^2\cdot 227$, which can be split into two factors in 3 ways, $2043\cdot 1=3\cdot 681=227\cdot 9$. This gives us three pairs of equations to solve for $n$: $n+m+6=2043$ and $n-m+6=1$ give $2n+12=2044$ and $n=1016$. $n+m+6=681$ and $n-m+6=3$ give $2n+12=684$ and $n=336$. $n+m+6=227$ and $n-m+6=9$ give $2n+12=236$ and $n=112$. Finally, $1016+336+112=1464$, so the answer is $\boxed{464}$.