According to an answer in [this question](https://mathoverflow.net/questions/343876/rational-diophantine-set-for-the-non-squares?noredirect=1#comment859959_343876) the set of integer non-squares is diophantine over the rationals: 
there is polynomial $P(a,x_1,...,x_n)$ which has rational solution
iff $a$ is integer non-square. For the variety take $P^2+(x_a^2-a)^2$.