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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 suchmodel correctly reproduces the UV luminosity function at $z>10$, and the starformation rate (SFR) density evolution. The model also predicts, in agreementwith 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 $\rmsSFR^\star = 25\, \rm Gyr^{-1}$ and $\approx 45$% of the galaxies becomesuper-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 theoutflow, respectively. This prediction is consistent with the outflowvelocities measured in 12 super-Eddington galaxies of the JWST/JADES sample.Such outflows clear the dust, thus boosting the galaxy luminosity. They alsodramatically enhance the visibility of the Ly$\alpha$ line from $z>10$galaxies, by introducing a velocity offset. The observed Ly$\alpha$ propertiesin GN-z11 ($z=10.6$) are simultaneously recovered by the outflow model if $\logN_{\rm HI} \simeq 20.1$, implying that the outflow is largely ionized. We makeanalogous predictions for the Ly$\alpha$ visibility of other super-earlygalaxies, and compare the model with Ly$\alpha$ surveys at $z>7$, finding thatessentially all super-Eddington (sub-Eddington) galaxies are (not) detected inLy$\alpha$. Finally, the sSFR positively correlates with the LyC escapefraction as outflows carve ionized, transparent channels through which LyCphotons leak.

Super-early JWST galaxies, outflows and Lyman alpha visibility in the EoR