We use high-resolution cosmological zoom-in simulations in order to analyze galaxy evolution at redshifts z~6-12 in highly-overdense 5 sigma density peaks. Strong stellar feedback, in the form of galactic winds, is expected to play an important role in the evolution of these regions. We investigate the effects of these winds by comparing different galactic outflow prescriptions, including (i) a constant velocity model (CW), (ii) a variable velocity model scaling with galaxy properties (VW), and (iii) a model with no outflows (NW). The CW model is also applied to a simulation of an average density region to study the impact of environment on galaxy evolution. A direct consequence of the overdensity is a shallow galaxy mass function slope at the low-mass end and an accelerated evolution of dark matter and baryonic structures. The overdensity hosts massive haloes, up to ~10^{12} Msun, with embedded galaxies up to ~10^{11} Msun in stellar mass by z~6, which are absent in the "normal" region. The CW model leads to similar gas fractions, star formation rates (SFRs) and metallicity in galaxies, in both environments due to the absence of scaling between wind and galaxy properties. Only the VW model is able to reproduce both the observed specific SFR (sSFR) evolution between z~8 and z~6 and the sSFR-stellar mass relation at z~6. The models also differ on the state of the intergalactic medium (IGM). Hot ~10^{4.5}-10^7 K and high-metallicity 0.03-0.1 Zsun gas fills up almost entirely the computational box for the CW model, while it remains confined in massive filaments for the VW case, and is locked up in galaxies for the NW case. Such gas is also nearly absent in the average density region. However, further constraints on the state of the IGM at high-z are needed to separate the models, as current estimates of metal-enrichment at z~5.7 are compatible with all our simulation results.