Maximize and Minimize?

A square $ABCD$ of unit area is given.

A quadrilateral $EFGH$ is drawn such that $E$ lies on $AB$ , $F$ lies on $BC$ , $G$ lies on $CD$ and $H$ lies on $DA$ .

If the minimum and maximum possible values for the sum of the squares of the lengths of the sides of $EFGH$ are $\alpha$ and $\beta$ respectively, find the value of:

$\alpha^{\beta} + \beta^{\alpha}$