  No-scale supersymmetry or gaugino mediation augmented with large negative
Higgs soft masses at the input scale provides a simple solution to the
supersymmetric flavor problem while giving rise to a neutralino LSP. However,
to obtain a neutralino LSP it is often necessary to have tachyonic input Higgs
soft masses that can give rise to charge-and-color-breaking (CCB) minima and
unbounded-from-below (UFB) directions in the low energy theory. We investigate
the vacuum structure in these theories to determine when such problematic
features are present. When the standard electroweak vacuum is only metastable,
we compute its lifetime under vacuum tunneling. We find that vacuum
metastability leads to severe restrictions on the parameter space for larger
$\tan\beta \sim 30$, while for smaller $\tan\beta\sim 10$, only minor
restrictions are found. Along the way, we derive an exact bounce solution for
tunneling through an inverted parabolic potential.
