  We discuss the implications of assuming a four-zero Yukawa texture and a
general Higgs potential for the production of neutral Higgs boson pairs at
\gamma \gamma colliders through the \gamma \gamma \to \phi_i\phi_j (\phi_i = h,
H, A) reaction within the context of the two Higgs doublet model type III.
Exact analytical expressions for the \gamma \gamma \to \phi_i\phi_j reaction
are presented. The use of a nonlinear R_\xi-gauge, which considerably
simplifies the loop calculations and renders compact analytical expressions, is
stressed. We show that these processes are very sensitive to a general
structure of the Higgs potential that impact the triple and quartic couplings
of the scalar sector. We present results for scenarios of the parameters of the
model that are still consistent with current experimental constraints. It is
found that the cross sections for the \gamma \gamma \to \phi_i\phi_j processes
can be up to two orders of magnitude larger than those gotten in 2HDM type I
and type II. The possibility of a light CP--scalar is also studied.
