Deviations from the predictions of general relativity due to energy-momentum squared gravity (EMSG) are expected to become pronounced in the high density cores of neutron stars. We derive the hydrostatic equilibrium equations in EMSG and solve them numerically to obtain the neutron star mass-radius relations for four different realistic equations of state. We use the existing observational measurements of the masses and radii of neutron stars to constrain the free parameter, alpha, that characterizes the coupling between matter and spacetime in EMSG. We show that -10(-38) cm(3)/erg < alpha < +10(-37) cm(3)/erg. Under this constraint, we discuss what contributions EMSG can provide to the physics of neutron stars, in particular, their relevance to the so called hyperon puzzle in neutron stars. We also discuss how EMSG alters the dynamics of the early universe from the predictions of the standard cosmological model. We show that EMSG leaves the standard cosmology safely unaltered back to t similar to 10(-4) seconds at which the energy density of the universe is similar to 10(34) erg cm(-3).