Let $p > 3$ be prime. Is is true that there exists $x \in \mathbb{Z}_p$ such that 
$$
(1+x^2)^3-1
$$
is not a square in $\mathbb{Z}_p$? Equivalently, can we show that the equation
$$
-y^2 = (1+x^2)^3-1
$$
always has non-trivial solutions $(x,y) \ne (0,0)$ in $\mathbb{Z}_p$?