  We study the cosmology of the Supersymmetric Standard Model augmented by a
gauge singlet to solve the $\mu$-problem and describe the evolution of the
domain walls which are created during electroweak symmetry breaking due to the
discrete $Z_{3}$ symmetry in this model. The usual assumption, that
non-renormalizable terms induced by gravity (which explicitly break this
symmetry) may cause the walls to collapse on a cosmologically safe timescale,
is reconsidered. Such terms are constrained by considerations of primordial
nucleosynthesis, and also by the fact that by not respecting the $Z_{3}$
symmetry they induce divergences which destabilise the hierarchy and
reintroduce the $\mu$--problem. We find that, even when the K\"ahler potential
is `non-minimal' (i.e. when the hidden sector couples directly to the visible)
the model is either ruled out cosmologically or suffers from a naturalness
problem.
