The set $A$ of non-squares (of rationals) is Diophantine in $\mathbb{Q}$ by [1]. The set $B:=\mathbb{Q}\smallsetminus\mathbb{Z}$ is also Diophantine by [2]. The set of non-squares of integers is equal to $A\cup B$, hence Diophantine.

For a generalization of [1], see also [3].

**[EDIT]** The paper [1] treats arbitrary (non-)$n$-th powers, but the case of (non-)squares was proved earlier by Poonen [4].

[1] *Colliot-Thélène, Jean-Louis; van Geel, Jan*, **Le complémentaire des puissances $n$-ièmes dans un corps de nombres est un ensemble diophantien**, Compos. Math. 151, No. 10, 1965-1980 (2015). ZBL1346.14066..

[2] *Koenigsmann, Jochen*, **Defining $\mathbb Z$ in $\mathbb Q$**, Ann. Math. (2) 183, No. 1, 73-93 (2016). ZBL1390.03032..

[3] *Dittmann, Philip*, **Irreducibility of polynomials over global fields is diophantine**, Compos. Math. 154, 761-772 (2018). ZBL06861881.

[4]
*Poonen, Bjorn*, **The set of nonsquares in a number field is diophantine**, Math. Res. Lett. 16, No. 1, 165-170 (2009). ZBL1183.14031.