[en] We studied the effect of physical pressure on the electronic and magnetic properties of ferrimagnetic double perovskites A₂FeReO₆ (A = Ca, Ba) using Re L_2,3 edge x-ray absorption spectroscopy and powder diffraction measurements. Volume compression is shown to dramatically increase the magnetic coercivity (H_c) in polycrystalline samples of both compounds with ΔH_c/ΔV similar to 150-200 Oe/ų. A nearly eight-fold increase in H_c, from 0.2 to 1.55 T, is obtained in Ba₂FeReO₆ at P = 29 GPa. While no signs of structural phase transitions are seen in either sample to similar to 30 GPa, the structural data points to a pressure-driven increase in tetragonal distortion of ReO₆ octahedra. A sizable but pressure-independent Re orbital-to-spin magnetic moment ratio is observed, pointing to the critical role of spin-orbit interactions at Re sites. We present a J(eff) description of the electronic structure that combines effects of crystal field and spin-orbit coupling on the Re 5d² orbitals and use this description to provide insight into the pressure-induced enhancement of magnetic anisotropy.