= Solution
Subtract $M_0$, which does not affect either the integral or its <quadratic variation>. Let $N=M-M_0$. Choose a <localizing sequence> $\tau_n$ for $N$. By local boundedness of the <previsible process> $H$, there are increasing stopping times $\rho_n\uparrow\infty$ and finite deterministic constants $K_n$ with $|H_s|\leq K_n$ on $0<s\leq\rho_n$. Put
$$
\sigma_n=\tau_n\wedge\rho_n\wedge n\wedge\inf\{t:|N_t|\geq n\text{ or }[M]_t\geq n\}.
$$
Continuity and finiteness on compact intervals imply $\sigma_n\uparrow\infty$ almost surely. The stopped process $N^{\sigma_n}$ is a bounded square-integrable martingale, its bracket is bounded by $n$, and its stopped integrand is bounded by $K_n$. The indicator $\mathbf1_{\{s\leq\sigma_n\}}$ is predictable, so the stopped stochastic integral identity is valid.
Fix $n$ and abbreviate
$$
J_t=\int_0^{t\wedge\sigma_n}H_s\,dM_s,\qquad Q_t=\int_0^{t\wedge\sigma_n}H_s^2\,d[M]_s.
$$
The <Itô isometry> gives $\mathbb E J_t^2=\mathbb E Q_t\leq K_n^2n$, so $J$ is a square-integrable martingale. For $F\in\mathcal F_s$, apply the same isometry to the predictable integrand $\mathbf1_F\mathbf1_{(s,t]}H$, stopped at $\sigma_n$. This yields
$$
\mathbb E\bigl[\mathbf1_F(J_t-J_s)^2\bigr]=\mathbb E\bigl[\mathbf1_F(Q_t-Q_s)\bigr].
$$
Also $\mathbb E[J_t-J_s\mid\mathcal F_s]=0$. Expanding $J_t^2-J_s^2$ therefore proves that $J^2-Q$ is a true martingale. The process $Q$ is continuous adapted increasing and starts at zero. By the characterization of continuous martingale <quadratic variation> as its increasing square compensator, and <uniqueness of an increasing square compensator>, we have $[J]=Q$.
Stochastic integration and quadratic variation commute with stopping; for the latter this follows either from squared-increment sums or from the same square-compensator characterization. Hence the desired identity holds up to every $\sigma_n$. Since these times increase to infinity, they exhaust every finite time interval almost surely, and the continuous identities patch to give
$$
\boxed{[H\mathbin\cdot M]_t=\int_0^tH_s^2\,d[M]_s.}
$$
This is the <localized isometry proof of stochastic-integral quadratic variation>. No unstopped second-moment or bracket-integrability assumption has been made.
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