The perovskite antiferromagnetic ($T_{\rm N}$ $∼$ 220 K) insulator EuNiO$_3$ undergoes at ambient pressure a metal-to-insulator transition at $T_{\rm MI}$ = 460 K which is associated with a simultaneous orthorhombic-to-monoclinic distortion, leading to charge disproportionation. We have investigated the change of the structural and magnetic properties of EuNiO$_3$ with pressure (up to $∼$ 20 GPa) across its quantum critical point (QCP) using low-temperature synchrotron angle-resolved x-ray diffraction and $^{151}$Eu nuclear forward scattering of synchrotron radiation, respectively. With increasing pressure we find that after a small increase of $T_{\rm N}$ ($p$ $≤$ 2 GPa) and the induced magnetic hyperfine field $B_{\rm hf}$ at the $^{151}$Eu nucleus ($p$ $≤$ 9.7 GPa), both $T_{\rm N}$ and $B_{\rm hf}$ are strongly reduced and finally disappear at $p_{\rm c}$ $≅$ 10.5 GPa, indicating a magnetic QCP at $p_{\rm c}$. The analysis of the structural parameters up to 10.5 GPa reveals no change of the lattice symmetry within the experimental resolution. Since the pressure-induced insulator-to-metal transition occurs at $p_{\rm IM}$ $≅$ 6 GPa, this result implies the existence of an antiferromagnetic metallic state between 6 and 10.5 GPa. We further show from the analysis of the reported high pressure electrical resistance data on EuNiO$_3$ at low-temperatures that in the vicinity of the QCP the system behaves as non-Fermi-liquid, with the resistance changing as $T^{\rm n}$, with n=1.6, whereas it becomes a normal Fermi-liquid, n = 2, for pressures above $∼$15 GPa. On the basis of the obtained data a magnetic phase diagram in the ($p$, $T$) space is suggested. ; Comment: accepted in Physical Review B