In contrast to hafnia (HfO2) thin films, where the appearance of switchable ferroelectric polarization can be induced by strain or defect engineering, reliable methods for controlling ferroelectricity in HfO2 nanoparticles are absent. Direct experimental observations of ferroelectric hysteresis and ferroelectric domains in these nanoparticles have also not been reported. To the best of our knowledge, stress-induced ferroelectric states in the HfO2 nanoparticles have not been explored. In this work, we study the influence of chemical stress on phase diagrams, dielectric properties, and polar properties of spherical HfO2 core-shell nanoparticles using a Landau-Ginzburg-Devonshire free energy functional that includes trilinear and biquadratic couplings involving polar, antipolar, and nonpolar order parameters. The ferroelectric phase exhibits reentrant behavior as a function of nanoparticle size, such that the spontaneous polarization exists only within a limited range of core radii R_c, namely R_cr^min<R_c<R_crmax. The minimal critical radius R_cr^min is primarily determined by the size dependence of the depolarization field and correlation effects, whereas the maximal critical radius R_cr^max is governed mainly by the size dependence of chemical stresses induced by the elastic defects in the shell. Thus, this work identifies a stress-driven mechanism for reentrant ferroelectricity stabilization in nanoscale hafnia systems, arising from the competition between depolarization field-induced suppression of ferroelectricity and its stabilization by shell-induced chemical stress. We revealed that relatively large compressive chemical strains are necessary to induce the ferroelectric phase in the HfO2 nanoparticles. Successful chemical strain engineering opens the way for significant enhancement of nanoscale HfO2 polar properties for applications in advanced memory cells and logic devices.