  In this paper, we study the Jeans analysis in the context of
energy-momentum-squared gravity (EMSG). More specifically we find the new Jeans
mass for non-rotating infinite mediums as the smallest mass scale for local
perturbations that can be stable against its own gravity. Furthermore, for
rotating mediums, specifically for rotating thin disks in the context of EMSG,
we find a new Toomre-like criterion for the local gravitational stability.
Finally, the results are applied to a hyper-massive neutron star, as an
astrophysical system. Using a simplified toy model we have shown that, for a
positive (negative) value of the EMSG parameter $\alpha$, the system is stable
(unstable) in a wide range of $\alpha$. On the other hand, no observational
evidence has been reported on the existence of local fragmentation in HMNS.
Naturally, this means that EMSG with positive $\alpha$ is more acceptable from
the physical point of view.
