  We study the production of the SM Higgs boson (H^0) at future e gamma
colliders, through the reaction e gamma --> e H^0. The amplitude is evaluated
using the non-linear R_\xi gauge, which greatly simplifies the calculation.
Complete analytical expressions for the amplitudes are presented, which include
the contributions from 1-loop triangles "gamma gamma^* H^0" and "gamma Z^* H^0"
as well as the W- and Z-boxes with their related $eeH^0$ triangle graphs. The
resulting cross section for this mechanism indicates that it could be used to
detect the Higgs signal and to test its properties.
