Can $(X_n)_{n \ge 0}$ and $(X_n^2)_{n \ge 0}$ be both martingales?
Your argument, as Kurt pointed out, is correct.
An other way to see this is to define $D_n=X_{n}-X_{n-1}$ and $D_0=0$. Then $$ \mathbb E\left[X_n^2\mid\mathcal F_{n-1}\right]=\mathbb E\left[D_n^2\mid\mathcal F_{n-1}\right]+X_{n-1}^2 $$ hence $X_n^2$ will be a martingale if and only if $\mathbb E\left[D_n^2\mid\mathcal F_{n-1}\right]=0$. Taking $\Omega$ in the definition of conditional expectation gives that $D_n=0$ almost surely hence the only possibility is a constant martingale.