When n=2.

The answer is all positive integer $n\geq3$ For $n=1$, Let $P(x)=x+c\quad; c \in\mathbb{Z}$ Notice that $P(-c)P(-c)=P(-c).$ So, this case clearly fails. For $n=2$, Let $P(x)=x^2+mx+n\quad; m,n \in \mathbb{Z}$ Notice that $P(n)=P(1)P(0).$ So, this case also fails. For $n\geq3$, Construct a polynomial $P(x)=x^{n-2}(x-1)(x-2)+2$ $\Rightarrow P(x)\equiv 2 \pmod{3}\quad\forall x\in \mathbb{Z}$ $\Rightarrow P(a)P(b)\neq P(c)\quad \forall a,b,c \in \mathbb{Z}\quad\blacksquare$