  We show that the Higgs production in $e\gamma$ collisions, through the
one-loop reaction $e\gamma\to eH$ at large transverse momentum, can provide a
precise determination of the $Z\gamma H$ vertex. We concentrate on the analysis
of signal rates vs background looking for strategies to improve the ratio
$S/B$. Possible bounds on anomalous contributions to the $Z\gamma H$ vertex are
derived.
