  The constraints on the anomalous quartic $WW\gamma\gamma$ and
$ZZ\gamma\gamma$ gauge boson couplings are investigated through the processes
$e\gamma\to W^{-}\gamma\nu_{e}$ and $e\gamma\to Z\gamma e$. Considering the
longitudinal and transverse polarization states of the final W or Z boson and
incoming beam polarizations we find 95% confidence level limits on the
anomalous coupling parameters $a_{0}$ and $a_{c}$ with an integrated luminosity
of 500 $fb^{-1}$ and $\sqrt{s}$=0.5, 1 TeV energies. Assuming the
$W^{+}W^{-}\gamma\gamma$ couplings are independent of the $ZZ\gamma\gamma$
couplings we show that the longitudinal polarization state of the final gauge
boson improves the sensitivity to anomalous couplings by a factor of 2-3
depending on energy and coupling. An extra enhancement in sensitivity by a
factor of 1.3 comes from a set of initial beam polarizations.
