  Motivated by the recent findings that the ratio $\varepsilon'/\varepsilon$ in
the Standard Model (SM) appears to be significantly below the data we
investigate whether the necessary enhancement of this ratio can be obtained in
331 models, in which new physics (NP) contributions to
$\varepsilon'/\varepsilon$ and other flavour observables are dominated by
tree-level exchanges of a $Z^\prime$. NP contributions to
$\varepsilon'/\varepsilon$ in these models are governed by the electroweak
operator $Q_8$ for which the hadronic matrix element is rather well known so
that our analysis of NP contributions is subject to much smaller uncertainties
than within the SM. We consider seven 331 models selected in our earlier
analysis on the basis of electroweak precision data. Imposing the constraints
from $\Delta F=2$ transitions we find that only three of these models can
provide a significant positive shift in $\varepsilon'/\varepsilon$ up to
$6\times 10^{-4}$ for $M_{Z^\prime}=3$ TeV. Two of them allow simultaneously a
supression of ${\cal B}(B_{s}\to \mu^+\mu^-)$ by $20\%$, bringing the theory
closer to the data without any significant impact on the Wilson coefficient
$C_9$. The third one provides also the shift $\Delta C_9=-0.6$, softening the
anomalies in $B\to K^*\mu^+\mu^-$, without any significant impact on $B_{s}\to
\mu^+\mu^-$. NP effects in rare $K$ decays and in $B\to K(K^*)\nu\bar\nu$ turn
out to be small. The flavour structure of 331 models implies that even for
$M_{Z^\prime}=30$ Tev a shift of $\varepsilon'/\varepsilon$ up to $8\times
10^{-4}$ and a significant shift in $\varepsilon_K$ can be obtained, while the
effects in other flavour observables are small.
