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It looks like $Z(t)$ is dependent on $t$ through $X_t$ and not directly on $t$, loosely speaking. Assume that $Z_t = f(X_t)$ and use Ito's formula with just one variable. $$dZ_t = \frac{df}{dX} dX_t + \frac{1}{2} \frac{d^2f}{dX^2} d[X_t,X_t]$$ Can you proceed and finish it?
You do not really need the dynamics of $S_t^2$. You can simply apply your standard technique from risk-neutral pricing. The time zero price of a European-style contract with payoff $X$ is given by $$V_0=e^{-rT}\mathbb{E}^\mathbb{Q}[X\mid\mathcal{F}_0].$$ Thus, \begin{align*} V_0 &= e^{-rT}\mathbb{E}^\mathbb{Q}[\mathbb{1}_{\{S_T^2\geq K\}}] \\ &= e^{-...