  One-loop production of a Higgs boson in electro-photon collisions at future
accelerators is studied via the process (e gamma -> e H), for intermediate
Higgs masses. Exact cross sections, including the possibility of longitudinally
polarized initial beams, are presented. Confirming previous estimates made in
the Weizs\"acker-Williams approximation, they are found to be more than two
orders of magnitude larger than the cross sections for the crossed process (e
e-> gamma H), in the energy range $\sqrt{s}=(0.5-2)$ TeV. We show that, not
only (e gamma -> e H) has a similar potential as the (gamma gamma -> H) process
for testing the one-loop gamma-gamma-H vertex, but, by requiring a final
electron tagged at large angle in (e gamma -> e H), the (H e) production
provides an excellent way of testing the Z-gamma-H vertex, too. Kinematical
distributions for the (e gamma -> e H -> e bb) process with a tagged final
electron are analyzed, and strategies for controlling the main irreducible
background are found. Initial-state-radiation effects are checked to be within
a few percent.
