The expansion of $ (mx+n)^2 $ is $ m^2x^2+2mnx+n^2 $. If $$16x^2+36x+56 = m^2x^2+2mnx+n^2,$$we can equate the coefficients of $ x^2 $ to get $ 16=m^2 $, and thus $ m=4 $ (we ignore the solution $ -4 $ because we are given that $ m $ is positive). We can also equate the coefficients of $ x $ above to get $ 36=2mn=2(4)n=8n $, and thus $ n=\frac{36}{8}=\frac{9}{2} $. So, $ mn=4\cdot\frac92 = \boxed{18} $.