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The James Webb Space Telescope (JWST) will provide deep imaging and spectroscopy for sources at redshifts above 6, covering the entire Epoch of Reionization (EoR, 6 <  z <  10), and enabling the detailed exploration of the nature of the different sources during the first 1 Gyr of the history of the Universe. The Medium Resolution Spectrograph (MRS) of the mid-IR Instrument (MIRI) will be the only instrument on board JWST able to observe the brightest optical emission lines Hα and [OIII]0.5007 μm at redshifts above 7 and 9, respectively, providing key insights into the physical properties of sources during the early phases of the EoR. This paper presents a study of the Hα fluxes predicted by state-of-the-art FIRSTLIGHT cosmological simulations for galaxies at redshifts of 6.5–10.5, and its detectability with MIRI. Deep (40 ks) spectroscopic integrations with MRS will be able to detect (signal-to-noise ratio > 5) EoR sources at redshifts above 7 with intrinsic star formation rates (SFR) of more than 2 M⊙ yr−1, and stellar masses above 4–9 × 107M⊙. These limits cover the upper end of the SFR and stellar mass distribution at those redshifts, representing ∼6% and ∼1% of the predicted FIRSTLIGHT population at the 6.5–7.5 and 7.5–8.5 redshift ranges, respectively. In addition, the paper presents realistic MRS simulated observations of the expected rest-frame optical and near-infrared spectra for some spectroscopically confirmed EoR sources recently detected by ALMA as [OIII]88 μm emitters. The MRS simulated spectra cover a wide range of low metallicities from about 0.2–0.02 Z⊙, and different [OIII]88 μm/[OIII]0.5007 μm line ratios. The simulated 10 ks MRS spectra show S/N in the range of 5–90 for Hβ, [OIII]0.4959,0.5007 μm, Hα and HeI1.083 μm emission lines of the currently highest spectroscopically confirmed EoR (lensed) source MACS1149-JD1 at a redshift of 9.11, independent of metallicity. In addition, deep 40 ksec simulated spectra of the luminous merger candidate B14-65666 at 7.15 shows the MRS capabilities of detecting, or putting strong upper limits on, the weak [NII]0.6584 μm, [SII]0.6717,0.6731 μm, and [SIII]0.9069,0.9532 μm emission lines. These observations will provide the opportunity of deriving accurate metallicities in bright EoR sources using the full range of rest-frame optical emission lines up to 1 μm. In summary, MRS will enable the detailed study of key physical properties such as internal extinction, instantaneous star formation, hardness of the ionizing continuum, and metallicity in bright (intrinsic or lensed) EoR sources. [ABSTRACT FROM AUTHOR]