  We study the processes $\gamma \gamma \to K^+ K^- \eta$ and $\gamma \gamma
\to K^+ K^- \pi^0$ using a data sample of 519 $fb^{-1}$ recorded with the BaBar
detector operating at the SLAC PEP-II asymmetric-energy $e^+ e^-$ collider at
center-of-mass energies at and near the $\Upsilon(nS)$ ($n = 2,3,4$)
resonances. We observe $\eta_c \to K^+ K^- \eta$ and $\eta_c \to K^+ K^- \pi^0$
decays, measure their relative branching fraction, and perform a Dalitz plot
analysis for each decay. We observe the $K^*_0(1430) \to K \eta$ decay and
measure its branching fraction relative to the $K \pi$ decay mode to be ${\cal
R}(K^*_0(1430)) = \frac{{\cal B}(K^*_0(1430) \to K \eta)}{{\cal B}(K^*_0(1430)
\to K \pi)} = 0.092 \pm 0.025^{+0.010}_{-0.025}$. The $\eta_c \to K^+ K^- \eta$
and $K^*_0(1430) \to K \eta$ results correspond to the first observations of
these channels. The data also show evidence for $\eta_c(2S) \to K^+ K^- \pi^0$
and first evidence for $\eta_c(2S) \to K^+ K^- \eta$.
