The notion of a map $f$ such that every pushout of $f$ is a weak equivalence has been considered in the recent preprint Homotopy theory for algebras over polynomial monads by Michael Batanin and Clemens Berger. They refer to such a map $f$ as a *couniversal weak equivalence.* They also introduce the notion of an $h$-cofibration as a map $g:A\to B$ such that in any diagram as follows where both squares are pushout squares then the map $w':X'\to Y'$ is a weak equivalence as soon as $w:X\to Y$ is:

\begin{array}{} A & \to & X & \to & Y \\ \downarrow & & \downarrow & & \downarrow
\\ B & \to & X' & \to & Y'\end{array}

Say $g$ is trivial if it's additionally a weak equivalence. The authors prove in Lemma 1.6 that every couniversal weak equivalence is a trivial $h$-cofibration. In a left proper model category couniversal weak equivalences are precisely trivial $h$-cofibrations. A further characterization of such maps in left proper model categories can be found in Lemma 1.5, including an equivalent definition which I independently introduced in my thesis (namely: every pushout square with $g$ as one of the legs is a homotopy pushout square). Properties of these maps can be found throughout section 1 of the paper and also in the last section of my paper on monoidal Bousfield localizations.

What you're asking is whether every trivial $h$-cofibration is a trivial cofibration. I doubt this is true in general or even in a setting where all objects are cofibrant. However, it might still be true in your setting. In Top, the $h$-cofibrations are like the closed neighborhood deformation retracts. In general it's hard to explicitly describe the class of $h$-cofibrations, but the relationship between $h$-cofibrations and cofibrations in Top gives some hope that what you're asking might be true.