  The sensitivity of the rare decays $\eta^{(\prime)}\to\pi^{0}\gamma\gamma$
and $\eta^{\prime}\to\eta\gamma\gamma$ to signatures of a leptophobic $B$ boson
in the MeV-GeV mass range is analyzed in this work. By adding an explicit
$B$-boson resonance exchange, $\eta\to B\gamma\to\pi^{0}\gamma\gamma$, to the
Standard Model contributions from vector and scalar meson exchanges, and
employing experimental data for the associated branching ratios, it allows us
to improve the current constraints on the $B$-boson mass $m_{B}$ and coupling
to Standard Model particles $\alpha_{B}$. From these constraints and the
analysis of the available experimental $\gamma\gamma$ invariant mass
distribution, we show that a $B$-boson signature in the resonant mass range
$m_{\pi^{0}}\lesssim m_{B}\lesssim m_{\eta}$ is strongly suppressed and would
be very difficult to experimentally identify, assuming that the leptophobic $B$
boson only decays to Standard Model particles. In contrast, the limits outside
this mass window are less stringent and the corresponding $t$- and $u$-channel
signatures may still be observable in the data, as it occurs with the
nonresonant Standard Model $\rho$, $\omega$ and $\phi$ meson exchanges. In
addition, we make use of experimental data from the
$\eta^{\prime}\to\pi^{0}\gamma\gamma$ and $\eta^{\prime}\to\eta\gamma\gamma$
decays to explore larger $B$-boson masses. Our results are relevant for the
$B$-boson search programs at existing and forthcoming light-meson facilities,
such as KLOE(-II) and Jefferson Lab Eta Factory experiments.
