Oil-well casing structures operating in deep underground environments are simultaneously subjected to high geomechanical stresses and elevated temperatures, which may significantly reduce their structural stability and service life. This study presents a unified analytical framework for evaluating the thermo-mechanical behavior of advanced oil-well casing systems using an equivalent graphene nanoplatelet-reinforced composite (GPLRC) sandwich beam with a re-entrant auxetic core. The proposed model is formulated based on Reddy’s third-order shear deformation theory, Hamilton’s principle, and the Navier solution to investigate the bending and buckling responses under coupled geomechanical and thermal loading conditions. Temperature-dependent material properties, effective in-situ geomechanical stresses, and thermally induced compressive forces are incorporated into the formulation to realistically represent downhole operating conditions. A comprehensive parametric study is conducted to examine the influences of GPL volume fraction, GPL distribution pattern, auxetic cell angle, geomechanical loading, and temperature on the structural performance of the equivalent casing model. The results demonstrate that increasing the GPL content from 0 to 1% reduces the mid-span deflection by approximately 68% and increases the critical buckling load by about 212%, while the FG-X distribution provides the best overall structural performance. The proposed configuration reaches the adopted linear-response limit at an effective external pressure of approximately 48.5 MPa, and thermo-mechanical buckling occurs at a critical temperature rise of about 242 °C for the highest investigated preload ratio. The findings demonstrate that the combined use of GPL-reinforced face sheets and a re-entrant auxetic core can significantly enhance the stiffness and global stability characteristics of the proposed equivalent sandwich-beam model. Furthermore, the developed analytical framework provides a computationally efficient reduced-order tool for preliminary parametric assessment of casing-related global response modes under representative high-pressure and high-temperature downhole conditions.