  The overabundance of super-early (redshift $z>10$), luminous ($M_{\rm UV} <
-20$), and blue galaxies detected by JWST has been explained (Ferrara et al.
2023) as due to negligible dust attenuation in these systems. We show that such
model correctly reproduces the UV luminosity function at $z>10$, and the star
formation rate (SFR) density evolution. The model also predicts, in agreement
with data, that the cosmic specific SFR grows as ${\rm sSFR} \propto
(1+z)^{3/2}$. At $z \simeq 10$ the cosmic sSFR crosses the critical value $\rm
sSFR^\star = 25\, \rm Gyr^{-1}$ and $\approx 45$% of the galaxies become
super-Eddington driving outflows reaching velocities of $\approx 830
\,(\epsilon_\star/f_M)^{1/2}\, {\rm km\, s}^{-1}$, where $\epsilon_\star$ and
$f_M$ are the SF efficiency and fraction of the halo gas expelled in the
outflow, respectively. This prediction is consistent with the outflow
velocities measured in 12 super-Eddington galaxies of the JWST/JADES sample.
Such outflows clear the dust, thus boosting the galaxy luminosity. They also
dramatically enhance the visibility of the Ly$\alpha$ line from $z>10$
galaxies, by introducing a velocity offset. The observed Ly$\alpha$ properties
in GN-z11 ($z=10.6$) are simultaneously recovered by the outflow model if $\log
N_{\rm HI} \simeq 20.1$, implying that the outflow is largely ionized. We make
analogous predictions for the Ly$\alpha$ visibility of other super-early
galaxies, and compare the model with Ly$\alpha$ surveys at $z>7$, finding that
essentially all super-Eddington (sub-Eddington) galaxies are (not) detected in
Ly$\alpha$. Finally, the sSFR positively correlates with the LyC escape
fraction as outflows carve ionized, transparent channels through which LyC
photons leak.
