  We study the production of gauge boson pairs at the next generation of linear
$e^+e^-$ colliders operating in the $e\gamma$ mode. The processes $e\gamma
\rightarrow VV^\prime F$ ($V,V^\prime =W$, $Z$, or $\gamma$ and $F=e$ or $\nu$)
can give valuable information on possible deviations of the quartic vector
boson couplings from the Standard Model predictions. We establish the range of
the new couplings that can be explored in these colliders based on a $3\sigma$
effect in the total cross section. We also present several kinematical
distributions of the final state particles that could manifest the underlying
new dynamics. Our results show that an $e\gamma$ collider can extend
considerably the bounds on anomalous interactions coming from oblique radiative
corrections and from direct searches in $e^+e^-$ colliders.
