  We study irradiated two-dimensional insulating bilayer honeycomb ferromagnets
and antiferromagnets coupled antiferromagnetically with a zero net
magnetization. The former is realized in the recently synthesized bilayer
honeycomb chromium triiodide CrI$_{\bf 3}$. In both systems, we show that
circularly-polarized electric field breaks time-reversal symmetry and induces a
dynamical Dzyaloshinskii-Moriya interaction in each honeycomb layer. However,
the resulting bilayer antiferromagnetic system still preserves a combination of
time-reversal and space-inversion ($\mathcal{PT}$) symmetry. We show that the
magnon topology of the bilayer antiferromagnetic system is characterized by a
$\pmb{\mathbb{Z}_2}$ Floquet topological invariant. Therefore, the system
realizes a magnonic Floquet quantum spin Hall insulator with spin filtered
magnon edge states. This leads to a non-vanishing Floquet magnon spin Nernst
effect, whereas the Floquet magnon thermal Hall effect vanishes due to
$\mathcal{PT}$ symmetry. We study the rich $\pmb{\mathbb{Z}_2}$ Floquet
topological magnon phase diagram of the system as a function of the light
amplitudes and polarizations. We further discuss the great impact of the
results on future experimental realizations.
