Let $(M_n)$ be a martingale with $M_n \in L^2$. $S,T$ are bounded stopping times w $S\leq T$. Show $M_T, M_T$ are both in $L^2$ and that
$E[(M_T-M_S)^2|\mathscr{F}_S]=E[M_T^2-M_S^2|\mathscr{F}_S]$
and that
$E[(M_T-M_S)^2]=E[M_T^2]-E[M_S^2]$
I believe I have shown the first equality, but I did not use $S \leq T$ or that M_T,M_S are in $L^2$, so I am concerned I made a mistake. My proof is:
$E[(M_T-M_S)^2|\mathscr{F}_S]=E(M_T^2|\mathscr{F}_S] + E[M_S^2|\mathscr{F}_S] - 2E[M_TM_S|\mathscr{F}_S]$, and then since $M_S$ is $\mathscr{F}_S$ measurable, $E[M_TM_S|\mathscr{F}_S]=M_SE[M_T|\mathscr{F}_S]=M_S^2=E[M_S^2|\mathscr{F}_S]$, which then proves the desired equality.
The second equality, I expanded everything out, but don't see how to proceed from there.
From your first equality, \begin{align*} E[(M_T - M_S)^2] &= E[E[(M_T - M_S)^2 | \mathcal{F}_S]] \\ &= E[E[M_T^2 - M_S^2|\mathcal{F}_S]] \\ &= E[M_T^2]-E[M_S^2]. \end{align*} You should also mention the martingales are right-continuous so that we may apply optional stopping.