Update
For each polynomial $P \in \mathbb{N}[X]$, let $S_P = \{ P(n) \mid n \in \mathbb{N}\}$. Does the Boolean algebra generated by the subsets $S_P$ of $\mathcal{P}(\mathbb{N})$ such that $P$ is a polynomial satisfying $P(0) = 0$ - contain a nonempty finite set?
Yes. The intersection of the sets of values of $x^2$ and $x^3+x$ is finite. This is because the elliptic curve $$ y^2=x^3+x $$ has only finitely many points with integer coordinates (IIRC this holds for all elliptic curves).
In this particular case we can also argue directly. As $x$ and $x^2+1$ are coprime, their product is a square, if and only if they are both squares. The latter is obviously a square only if $x=0$.
Thus $S_{x^2}\cap S_{x^3+x}=\{0\}$.
If $0\notin\Bbb{N}$ for you, then $y^2=x^3+4x$ should also work and give you $S_{x^2}\cap S_{x^3+4x}=\{(0),16\}.$