  We analyze the production and detection of the Higgs boson in the next
generation of linear $e^+e^-$ colliders operating in the $e\gamma$ mode. In
particular, we study the production mechanism $e + \gamma \rightarrow e \gamma
\gamma \rightarrow e + H$, where one photon is generated via the laser
backscattering mechanism, while the other is radiated via the usual
bremsstrahlung process. We show that this is the most important mechanism for
Higgs boson production in a $500$ GeV $e\gamma$ collider for
$M_H\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}140$ GeV. We also study
the signals and backgrounds for detection of the Higgs in the different decay
channels, $b \bar b$, $W^+W^-$, and $ZZ$, and suggest kinematical cuts to
improve the signature of an intermediate mass Higgs boson.
