According to an answer in [this question](https://mathoverflow.net/questions/343876/rational-diophantine-set-for-the-non-squares) the set of rational non-squares is diophantine over the rationals: 
there is polynomial $P(a,x_1,\dots,x_n)$ which for $a\in\mathbb{Q}$ has a rational solution
iff $a$ is a rational non-square. For the variety take the hypersurface with equation $P^2+(x_{n+1}^2-a)^2=0$ in $n+2$ coordinates $(a,x_1,\dots,x_{n+1}$).