A space $X$ is discretely generated (DG) if for every non-closed set $A \subset X$ and for every point $x \in \overline{A} \setminus A$ there is a discrete set $D \subset A$ such that $x \in \overline{D}$.
This is a convergence type property (and in fact, Fréchet-Urysohn and even radial spaces are DG) but it's pretty weak: every scattered space is DG, every compact space of countable tightness is DG in ZFC and every monotonically normal space is DG (see Dow, A.; Tkachenko, M.G.; Tkachuk, V.V.; Wilson, R.G., Topologies generated by discrete subspaces, Glas. Mat., III. Ser. 37, No.1, 187-210 (2002). ZBL1009.54005.).
There are examples of DG spaces with a non-DG square (see Murtinová, Eva, On products of discretely generated spaces, Topology Appl. 153, No. 18, 3402-3408 (2006). ZBL1107.54006.), but in a few notable special case discrete generability is indeed preserved in box products:
THEOREM (Tkachuk, 2012): Every box product of monotonically normal spaces is discretely generated.
THEOREM (Barriga-Acosta and Hernández-Hernández, 2016): Every box product of regular first-countable spaces is discretely generated.