Compression packers are widely used in oil and gas well operations for zonal isolation, yet the large-deformation mechanical behavior of their rubber sealing elements lacks a closed-form analytical solution. This paper presents a complete theoretical analysis of the axisymmetric compression of an annular rubber cylinder based on the incompressible Mooney–Rivlin hyper-elastic model. The deformation process is divided into three successive stages: free expansion, casing-constrained deformation, and fully constrained deformation. Analytical expressions for the principal stretches, stress fields, and axial force are derived for each stage by integrating the radial equilibrium equation with proper treatment of the Lagrange multiplier. The frictionless analytical results are validated against axisymmetric finite element simulations, showing excellent agreement for all stress components. The applicability of the frictionless theory to frictional conditions is then examined. Results show that although friction introduces non-uniform axial deformation and end bulging, the average contact pressure on the rubber–mandrel interface agrees closely with the theoretical prediction, especially at higher axial forces (150–250 kN). A linear relationship between the average contact pressure and the axial force is confirmed, providing a simple design tool. The theoretical solution offers a computationally efficient alternative to finite element analysis for preliminary packer design and parametric studies.