The evolution of shell structure and nuclear deformation in neutron-rich nuclei remains one of the major challenges in nuclear structure physics, particularly near regions associated with shell erosion and shape coexistence. In the present work, neutron-rich isotones with neutron numbers N=50 and N=34 are investigated within the Hartree–Fock (HF) plus Bardeen–Cooper–Schrieffer (BCS) framework using the Skyrme SLy5 parameterization. The potential energy surfaces (PES) are calculated as functions of the quadrupole deformation parameter β₂ in order to determine the equilibrium nuclear shapes and examine the evolution of shell stability in neutron-rich systems. The calculated results for ⁷⁰Ca show a nearly spherical minimum at β₂ = 0.001 together with a weak oblate configuration, while ⁷²Ti and ⁷⁴Cr exhibit multiple competing minima corresponding to spherical, oblate, and prolate shapes, indicating pronounced shape coexistence and weakening of the traditional N=50 shell closure. For the N=34 isotones, ⁵⁴Ca displays weak deformation with β₂ = −0.052, supporting enhanced spherical stability, whereas ⁵⁶Ti and ⁵⁸Cr exhibit increasing prolate deformation with β₂ = 0.101 and β₂ = 0.151, respectively, reflecting stronger collective effects and shell evolution. These results indicate that proton-neutron correlations, pairing interactions, and deformation-driving orbitals play a significant role in the evolution of shell structure and the emergence of shape coexistence phenomena in neutron-rich nuclei.