Let $\lambda \in \mathbb{R}, \lambda > 0$ and let $X, Y, Z \sim P(\lambda)$ (they have Poissons distribution) independent random variables..

For a Poisson variable, $E(X^2)=\lambda(\lambda+1)$, and not $\lambda$.

I can't follow your argument ! It's really unclear. Since $X,Y,Z$ are independent, indeed $$\mathbb E[XYZ]=\mathbb E[X]\mathbb E[Y]\mathbb E[Z]$$ and $$\mathbb E[(XYZ)^2]=\mathbb E\left[X^2\right]\mathbb E\left[Y^2\right]\mathbb E\left[Z^2\right].$$ Now, $$\mathbb E[X^2]=\mathbb E[Y^2]=\mathbb E[Z^2]=\lambda +\lambda ^2.$$ This because $$Var(X)=\lambda =\mathbb E[X^2]-\mathbb E[X]^2=\mathbb E[X^2]-\lambda ^2.$$ (same with $Y,Z$). I let you conclude.