  In this paper we realize the supersymmetric classical $W$-algebras
$\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ and
$\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ as differential algebras generated
by the coefficients of a monic superdifferential operator $L$. In the case of
$\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ (resp.
$\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$) this operator is even (resp.
odd). We show that the supersymmetric Poisson vertex algebra bracket on these
supersymmetric W-algebras is the supersymmetric analogue of the quadratic
Gelfand-Dickey bracket associated to the operator $L$. Finally, we construct
integrable hierarchies of evolutionary Hamiltonian PDEs on both W-algebras. A
key observation is that to construct these hierarchies on the algebra
$\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ one needs to introduce a new
concept of even supersymmetric Poisson vertex algebras.
