  Let $f$ be a positive definite (non-classic) integral quaternary quadratic
form. We say $f$ is strongly $s$-regular if it satisfies a regularity property
on the number of representations of squares of integers. In this article, we
prove that there are only finitely many strongly $s$-regular quaternary
quadratic forms up to isometry if the minimum of the nonzero squares that are
represented by the quadratic form is fixed. Furthermore, we show that there are
exactly $34$ strongly $s$-regular diagonal quaternary quadratic forms
representing one (see Table $1$). In particular, we use eta-quotients to prove
the strongly $s$-regularity of the quaternary quadratic form
$x^2+2y^2+3z^2+10w^2$, which is, in fact, of class number $2$ (see Lemma $5.5$
and Proposition $5.6$).
