OK, looks like I've got it. Since it is pretty late here now, it would be nice if someone could check that I haven't written some nonsense (I apologize in advance if I have).

Let $N(H)$ be the kernel of $H$, let $Q$ be the orthogonal projector to $N(H)$ and let $P$ be the orthogonal projector to $N(H)^\perp$. Note that since $H$ is real symmetric, we have $QH=HQ=0$ and $PHP=H$.

Now let $e$ be a unit vector and let $X=x-\langle Hx,e\rangle e$. Then $PX=Px-\langle PHP(Px),Pe\rangle Pe$. Thus, if $x_j\in \mathbb R^n$ satisfy the recursion $x_{j+1}=x_j-\langle Hx_j,e_j\rangle e_j$ where $e_j$ cycle through an orthonormal basis in $\mathbb R^n$, then $y_j=Px_j\in P(\mathbb R^n)$ satisfy the recursion $y_{j+1}=y_j-\langle \widetilde H y_j,v_j\rangle v_j$, where $v_j=Pe_j$ cycle through some complete system of vectors whose norms are bounded by $1$. Moreover, if $H$ has a negative eigenvalue, so does $\widetilde H$, but the advantage is that $\widetilde H$ is invertible in $P(\mathbb R^n)$.

Thus, we reduced the problem to the following:

Let $E$ be a real Euclidean space, let $v_j\in E$ be a sequence cycling through a fixed **complete** system of vectors with norm at most $1$, let $H:E\to E$ be any **invertible** symmetric linear operator of norm at most $1$, let $x_0\in E$ be any vector with $\langle Hx_0,x_0\rangle<0$. Then the sequence of vectors defined by the recursion $x_{j+1}=x_j-\langle Hx_j,v_j\rangle v_j$ grows at least geometrically in norm.

The key inequality is
$$
\langle Hx_{j+1},x_{j+1}\rangle = \langle Hx_{j},x_{j}\rangle-\langle Hx_{j},v_{j}\rangle^2(2-\langle Hv_{j},v_{j}\rangle)
\\
\le \langle Hx_{j},x_{j}\rangle-\langle Hx_{j},v_{j}\rangle^2\,.
$$
Since $H$ is invertible, we have $|HX|\ge c|X|$ for all $X\in E$ with some $c>0$. Now, if $|\langle Hx_j,v_j\rangle|<\delta|x_j|$ throughout the whole cycle with very small $\delta$, then all vectors in the cycle differ from the first vector $X$ in the cycle by at most $O(\delta)|X|$ and we derive that $|\langle HX,v_j\rangle|=O(\delta)|X|$, which, due to the completeness of the system of vectors $v_j$ over which we cycle, results in $|HX|=O(\delta)|X|$, contradicting the lower bound if $\delta>0$ is small enough. Thus, in each cycle we never increase $\langle Hx_j,x_j\rangle$ and have the inequality
$$
\langle Hx_{j+1},x_{j+1}\rangle \le \langle Hx_j,x_j\rangle-\delta^2|x_j|^2
\\
\le (1+\delta^2)\langle Hx_j,x_j\rangle
$$
at least once. That is the desired geometric growth. The end.