Adaptive filter for noise cancellation when measuring some input
I'm a little skeptical of your derivation and notation as it abstains from telling whether a Mean-Square or a Least-Squares metric of error is minimized, but rather just works on an instantaneous error at sample $n$. But I believe you eventually would devise an LMS or RLS filter, hence either of the two would happen.

So, based on your notation the adaptive system error is $\hat{x}(n)$, which is an estimate of the true signal $x(n)$,  and you have two correlated noises $x_N$ and $H(x_N)$ which are uncorrelated with the true signal. The standard noise cancelling setup.

Had you used the steepest descend based quadratic MSE surface minimization procedure, your instantaneous gradient on the error surface would be something like:

$$ \nabla J(w) = - E\{ e(n) \bar{x}[n] \} $$

where the expectation is due to stochastic MSE framework. In your notation this would become: 

$$ \nabla J(w) = - E\{ \hat{x}(n) \bar{x}_N[n] \} $$

Now, if the optimum condition is reached, then the error is minimized, and we can assume that $\hat{x}(n)$ converged to the true signal $x(n)$ and you may replace it. Remember this is an approximate conditions actually...

$$ \nabla J(w) = - E\{ x(n) \bar{x}_N[n] \} $$

and since your noise source is uncrorrelated with the true signal, this expectation is zero, yielding the zero gradient at the minimum point.

In your case, you don't specifically use expectations, so might be harder to show why the gradient would go to zero. Furthermore, as always, be aware that gradient would only theoretically go to zero. Any practical implementation will deviate from this depending on how unrealistic your assumptions were.