  Within the context of energy-momentum squared gravity (EMSG), where
non-linear matter contributions appear in the gravitational action, we derive
the modified TOV equations describing the hydrostatic equilibrium of charged
compact stars. We adopt two different choices for the matter Lagrangian density
($\mathcal{L}_m= p$ versus $\mathcal{L}_m= -\rho$) and investigate the impact
of each one on stellar structure. Furthermore, considering a charge profile
where the electric charge density $\rho_{\rm ch}$ is proportional to the
standard energy density $\rho$, we solve numerically the stellar structure
equations in order to obtain the mass-radius diagrams for the MIT bag model
equation of state (EoS). For $\mathcal{L}_m= p$ and given a specific value of
$\beta$ (including the uncharged case when $\beta= 0$), the maximum-mass values
increase (decrease) substantially as the gravity model parameter $\alpha$
becomes more negative (positive). However, for uncharged configurations and
considering $\mathcal{L}_m= -\rho$, our numerical results reveal that when we
increase $\alpha$ (from a negative value) the maximum mass first increases and
after reaching a maximum value it starts to decrease. Remarkably, this makes it
a less trivial behavior than that caused by the first choice when we take into
account the presence of electric charge ($\beta \neq 0$).
