= Solution
Put $S=H\mathbin\cdot M$ and $A_t=\int_0^tH_s^2d[M]_s$. The candidate $A$ is continuous, adapted, nondecreasing, starts at zero and has integrable terminal value. We prove $S^2-A$ is a <martingale>, rather than assuming the desired <quadratic variation of a stochastic integral>.
The assumed boundedness of $M$ also makes $M^2-[M]$ a true <martingale> on every finite horizon. Indeed, apply its local <martingale> identity at a <localizing sequence>; bounded stopped squares and the <Fatou lemma> show that $[M]_t$ is integrable. The stopped expressions are then dominated by $\sup_s|M_s|^2+[M]_t$, so the <dominated convergence theorem> gives the unrestricted conditional identity on that horizon.
First let $H$ be a bounded <simple predictable process>. On each interval with coefficient $h$, an increment of $S$ is $h(M_t-M_a)$. The defining <quadratic variation> identity for $M$, together with the <martingale> property of $M$, makes
$$
(M_t-M_a)^2-([M]_t-[M]_a)
$$
a <martingale> for $t\geq a$. Expanding $S_t^2-S_a^2$ gives a term $2S_ah(M_t-M_a)$ and the square term above; their <conditional expectations> yield the desired identity on that interval. Integrability follows because the variables are <square-integrable>; concatenating the finitely many intervals gives $S^2-A$ a <martingale>.
For general bounded <predictable> $H$, choose bounded <simple predictable processes> $H^n\to H$ in $L^2(M)$, which is possible since they generate the <predictable sigma-algebra>. Set $S^n=H^n\mathbin\cdot M$ and $A_t^n=\int_0^t(H_s^n)^2d[M]_s$. The <Itô isometry> and <Doob L2 maximal inequality> imply $S_t^n\to S_t$ in $L^2$; the <Cauchy-Schwarz inequality> gives $(S_t^n)^2\to S_t^2$ in $L^1$ and
$$
\mathbb E\sup_t|A_t^n-A_t|
\leq\|H^n-H\|_{L^2(M)}\bigl(\|H^n\|_{L^2(M)}+\|H\|_{L^2(M)}\bigr)\longrightarrow0.
$$
Pass the <conditional expectation> identity for $(S^n)^2-A^n$ to the $L^1$ limit. Thus $S^2-A$ is a <martingale>. Uniqueness in the defining <quadratic variation> property now gives
$$
\boxed{[H\mathbin\cdot M]_t=\int_0^tH_s^2\,d[M]_s.}
$$
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