jmc

By AM-GM, $$\begin{array}{rl}(ae{)}^2+(bf{)}^2+(cg{)}^2+(dh{)}^2& \ge 4\sqrt[4]{(ae{)}^2(bf{)}^2(cg{)}^2(dh{)}^2}\\ & =4\sqrt[4]{(abcdefgh{)}^2}\\ & =24.\end{array}$$Equality occurs when $(ae{)}^2=(bf{)}^2=(cg{)}^2=(dh{)}^2,$ $abcd=4,$ and $efgh=9.$ For example, we can take $a=b=c=d=\sqrt{2}$ and $e=f=g=h=\sqrt{3}.$ Hence, the minimum value is $\boxed{24}.$