  $B_s - \bar B_s$ mixing has been measured recently by D0 and CDF. The range
predicted in the standard model is consistent with data. However, the standard
model central values for $\Delta M_{B_s}$ and also $\Delta M_ {B_d}$ are away
from the data which may be indications of new physics. Using the observed
values of $\Delta M_{B_s}$ and $\Delta M_{B_d}$ we study general constraints on
flavor changing $Z^\prime$ interactions. In models with non-universal
$Z^\prime$ couplings we find that significant enhancements over the standard
model are still possible in the rare decay modes $B\to X_s \tau^+ \tau^-$,
$B\to X_s \nu \bar \nu$ and $B_s\to \tau^+ \tau^-$. Tree level $Z^\prime$
contributions to $K\to \pi \nu\bar \nu$ are now constrained to be very small,
but one loop effects can still enhance the standard model rate by a factor of
two.
