jmc

The line through points $B$ and $C$ has slope $\frac{-1-7}{7-(-1)}=-1$. Since $(7,-1)$ lies on the line, the line has equation $$y-(-1)=-1(x-7),$$or $y=-x+6$. Thus $m=-1$, $b=6$, and $m+b=-1+6=\boxed{5}$.