Summary: This paper is devoted to the evaluation of the quasi-static approximation of the hydro-dynamics of compact stars in the energy-momentum squared gravity. This theory allows for the presence of a scalar created by the stress-energy tensor as \(T^2 = T_{\gamma\lambda} T^{\gamma\lambda}\) in the generalized action of the theory. For this purpose, we investigate the anisotropic and dissipative fluid composition in axial and reflection symmetric geometry. To grasp the idea of the proposed approximation, a collection of invariant velocities is defined. As a result, the evolution of compact stars is presented in this approximation by examining the associated generalized field, dynamical, and scalar equations to elicit all possible outcomes. The generalized heat-transport equation is evaluated to study the thermodynamics of the chosen system. In the context of energy-momentum squared gravity, the role of structure scalars in the dynamics of the compact stars is also investigated. Furthermore, the proposed approximation is used to determine the significance of kinematical variables, generalized heat-fluxes, and structure scalars for the evolution of self-gravitating compact stars.