We present Ly alpha and UV-nebular emission line properties of bright Ly alpha emitters (LAEs) at z = 6-7 with a luminosity of log L-Ly alpha/[erg s(-1)] = 43-44 identified in the 21 deg(2) area of the SILVERRUSH early sample developed with the Subaru Hyper Suprime-Cam survey data. Our optical spectroscopy newly confirms 21 bright LAEs with clear Ly alpha emission, and contributes to making a spectroscopic sample of 96 LAEs at z = 6-7 in SILVERRUSH. From the spectroscopic sample, we select seven remarkable LAEs as bright as Himiko and CR7 objects, and perform deep Keck/MOSFIRE and Subaru/nuMOIRCS near-infrared spectroscopy reaching the 3 sigma flux limit of similar to 2 x 10(-18) erg s(-1) for the UV-nebular emission lines of He II lambda 1640, CIV lambda lambda 1548,1550, and OIII]lambda lambda 1661,1666. Except for one tentative detection of C (IV), we find no strong UV-nebular lines down to the flux limit, placing the upper limits of the rest-frame equivalent widths (EW0) of similar to 2-4 angstrom for C-IV, He-II, and O-III] lines. We also investigate the VLT/X-SHOOTER spectrum of CR7 whose 6 sigma detection of He (II) is claimed by Sobral et al. Although two individuals and the ESO archive service carefully reanalyzed the X-SHOOTER data that are used in the study of Sobral et al., no He II signal of CR7 is detected, supportive of weak UV-nebular lines of the bright LAEs even for CR7. The spectral properties of these bright LAEs are thus clearly different from those of faint dropouts at z similar to 7 that have strong UV-nebular lines shown in the various studies. Comparing these bright LAEs and the faint dropouts, we find anti-correlations between the UV-nebular line EW0 and the UV-continuum luminosity, which are similar to those found at z similar to 2-3.