  In this paper, we examine superspecial genus-2 curves $C: y^2 =
x(x-1)(x-\lambda)(x-\mu)(x-\nu)$ in odd characteristic $p$. As a main result,
we show that the difference between any two elements in
$\{0,1,\lambda,\mu,\nu\}$ is a square in $\mathbb{F}_{p^2}$. Moreover, we show
that $C$ is maximal or minimal over $\mathbb{F}_{p^2}$ without taking its
$\mathbb{F}_{p^2}$-form (we also give a criterion in terms of $p$ that tells
whether $C$ is maximal or minimal). As these applications, we study the
maximality of superspecial hyperelliptic curves of genus $3$ and $4$ whose
automorphism groups contain $\mathbb{Z}/2\mathbb{Z} \times
\mathbb{Z}/2\mathbb{Z}$.
