Transition-metal and rare-earth borides are of considerable interest due to their electronic, mechanical, and magnetic properties as well as their structural stability under extreme conditions. Here, we report on a series of high-pressure Raman and x-ray diffraction experiments on the cubic rare-earth hexaboride EuB6 to an ultrahigh pressure of 187 GPa in a diamond anvil cell. In EuB6, divalent europium ions occupy the corners of the cubic structure, which encloses a rigid boron-bonded cage. So far, no structural phase transitions have been reported, while the nanoindentation studies indicate amorphization in nanoscale shear bands during plastic deformation. Our x-ray diffraction studies have revealed that the ambient cubic phase of EuB6 shows broadening and splitting of diffraction peaks starting at 72 GPa and the broadening continuing to 187 GPa. The high-pressure phase is recovered on decompression, and the Raman spectroscopy of the recovered sample from 187 GPa shows a downward frequency shift and broadening of T2g, Eg, and A1g modes of boron octahedron. The density functional theory simulations of EuB6 at 100 GPa have identified five possible lowest energy crystal structures. The experimental x-ray diffraction data at high pressures is compared with the theoretical predictions and the role of structural distortions induced by shear stresses is also discussed.