Context. X-ray emission from quasars (QSOs) has been used to assess supermassive black hole accretion properties up to z ≈ 6. However, at z > 6 only ≈15 QSOs are covered by sensitive X-ray observations, preventing a statistically significant investigation of the X-ray properties of the QSO population in the first Gyr of the Universe. Aims. We present new Chandra observations of a sample of 10 z > 6 QSOs, selected to have virial black-hole mass estimates from Mg II line spectroscopy $ \left(\log\frac{M_{\mathrm{BH}}}{M_⊙}=8.5{-}9.6\right) $ . Adding archival X-ray data for an additional 15 z > 6 QSOs, we investigate the X-ray properties of the QSO population in the first Gyr of the Universe. In particular, we focus on the L_UV − L_X relation, which is traced by the α_ox parameter, and the shape of their X-ray spectra. Methods. We performed photometric analyses to derive estimates of the X-ray luminosities of our z > 6 QSOs, and thus their α_ox values and bolometric corrections (K_bol = L_bol/L_X). We compared the resulting α_ox and K_bol distributions with the results found for QSO samples at lower redshift, and ran several statistical tests to check for a possible evolution of the L_UV − L_X relation. Finally, we performed a basic X-ray spectral analysis of the brightest z > 6 QSOs to derive their individual photon indices, and joint spectral analysis of the whole sample to estimate the average photon index. Results. We detect seven of the new Chandra targets in at least one standard energy band, while two more are detected discarding energies E > 5 keV, where background dominates. We confirm a lack of significant evolution of α_ox with redshift, which extends the results from previous works up to z > 6 with a statistically significant QSO sample. Furthermore, we confirm the trend of an increasing bolometric correction with increasing luminosity found for QSOs at lower redshifts. The average power-law photon index of our sample ( ⟨Γ⟩ = 2.20_−0.34^+0.39 and ⟨Γ⟩ = 2.13_−0.13^+0.13 for sources with < 30 and > 30 net counts, respectively) is slightly steeper than, but still consistent with, typical QSOs at z = 1 − 6. Conclusions. All of these results indicate a lack of substantial evolution of the inner accretion-disk and hot-corona structure in QSOs from low redshift to z > 6. Our data hint at generally high Eddington ratios at z > 6.