  In this work we analyze traversable wormhole spacetimes in the framework of a
covariant generalization of Einstein's General Relativity known as
energy-momentum squared gravity, or $f\left(R,\mathcal T\right)$ gravity, where
$R$ is the Ricci scalar, $\mathcal T=T_{ab}T^{ab}$, and $T_{ab}$ is the
energy-momentum tensor. Considering a linear $f\left(R,\mathcal
T\right)=R+\gamma \mathcal T$ form, we show that a wide variety of wormhole
solutions for which the matter fields satisfy all the energy conditions, namely
the null, weak, strong and dominant energy conditions, exist in this framework,
without the necessity for a fine-tuning of the free parameters that describe
the model. Due to the complexity of the field equations these solutions are
obtained through an analytical recursive algorithm. A drawback of the solutions
obtained is that they are not naturally localized, and thus a matching with an
external vacuum is required. For that purpose, we derive the junction
conditions for the theory, and we prove that a matching between two spacetimes
must always be smooth, i.e., no thin-shells are allowed at the boundary.
Finally, we use these junction conditions to match the interior wormhole
spacetime to an exterior vacuum described by the Schwarzschild solution, thus
obtaining traversable localized static and spherically symmetric wormhole
solutions satisfying all energy conditions for the whole spacetime range. We
also prove that the methods outlined in this work can be straightforwardly
generalized to more complicated dependencies of the action in $\mathcal T$, as
long as crossed terms between $R$ and $\mathcal T$ are absent.
