One-loop production of a Higgs boson in e gamma collisions at future accelerators is studied via the process e gamma-->eH, for intermediate Higgs-boson masses. Exact cross sections, including the possibility of longitudinally polarized initial beams, are presented. Confirming previous estimates made in the Weizsacker-Williams approximation, they an found to be more than 2 orders of magnitude larger than the cross sections for the crossed process e(+)e(-)-->H gamma, in the energy range root s=(0.5-2) TeV. We show that, not only does e gamma-->eH have 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 a large angle in e gamma-eH, the He production provides an excellent way of testing the Z gamma H vertex, too. Kinematical distributions for the e gamma-->eH-->e(<b(b)over bar>) 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.
