In this system we Set $U_{n}$ to be an arbitrary RV with a finite second moment (i.e. $E[U_{n}^{2}]<\infty$). Since $e_{n}=X_{n}-U_{n}$ and $\hat{e}_{n}=\hat{X}_{n}-U_{n}$ then we have that $$\begin{align*} E[(X_{n}-\hat{X}_{n})^{2}]&=E[(X_{n}-U_{n}-(\hat{X}_{n}-U_{n}))^{2}]\\ &=E[(e_{n}-\hat{e}_{n})^{2}]\\ &=E[(e_{n}-Q(e_{n}))^{2}] \end{align*}$$Here we see the MSE of the reconstruction does not accumulate the error and is merely the MSE of the quantizer.