The James Webb Space Telescope will provide deep imaging and spectroscopy for sources at redshifts above 6, covering the Epoch of Reionization (EoR, 6 < z < 10). The Mid-IR instrument (MIRI) integral field spectrograph (MRS) will be the only instrument on board JWST able to observe the brightest optical emission lines H$alpha$ and [OIII]0.5007$mu$m at redshifts above 7 and 9, respectively. This paper presents a study of the H$alpha$ fluxes predicted by FIRSTLIGHT cosmological simulations for galaxies at redshifts of 6.5 to 10.5, and its detectability with MIRI. Deep (40 ks) spectroscopic integrations with MRS will be able to detect (S/N > 5) EoR sources at redshifts above 7 with intrinsic star formation rates of more than 2 M$_{odot}$ yr$^{-1}$, and stellar masses above 4-9 $times$ 10$^7$ M$_{odot}$. 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 detected by ALMA as [OIII]88$mu$m emitters. The MRS simulated spectra cover a wide range of low metallicities from about 0.2 to 0.02Z$_{odot}$, and different [OIII]88$mu$m/[OIII]0.5007$mu$m line ratios. The simulated 10ks MRS spectra show S/N in the range of 5 to 90 for H$beta$, [OIII]0.4959,0.5007$mu$m, H$alpha$ and HeI1.083$mu$m emission lines of MACS1149-JD1 at z = 9.11, independent of metallicity. In addition, deep 40 ks simulated spectra of the luminous merger candidate B14-65666 at z=7.15 shows the MRS capabilities of detecting, or putting strong upper limits, on the [NII]0.6584$mu$m, [SII]0.6717,0.6731$mu$m, and [SIII]0.9069,0.9532$mu$m emission lines. In summary, MRS will enable the detailed study of key physical properties like internal extinction, instantaneous star formation, hardness of the ionising continuum, and metallicity, in bright (intrinsic or lensed) EoR sources.
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