Positivity of GARCH-X models:
When it comes to GARCH models with exogenous regressors, it is more "tricky" to ensure positivity of the model, due to the unspecified model-dynamics of your exogenous regressor (moreover exogenous regressors can have drastically different characteristics). Without any model-specification on the exogenous regressor $X_{t-1}$, it is common to restrict the parameter-space of the model in order to ensure positivity.
For simplicity, let us vaguely define the GJR-GARCH(1,1)-X model with demeaned returns, $r_t$:
\begin{align*}
r_t \vert \mathcal{F}_{t-1} &= \varepsilon_t\\
\varepsilon_t &= \sigma_t \cdot z_t\\
\sigma^2_t &= \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 + \gamma I_{t-1} \varepsilon_{t-1}^2 + \nu X_{t-1},
\end{align*}
where $z_t \overset{iid}{\sim} D(0,1)$ is a standardized distribution and
$$I_{t-1} =\begin{cases}
1 & \text{if } \varepsilon_{t-1} < 0 \\
0 & \text{if } \varepsilon_{t-1} \geq 0
\end{cases}.$$
I have detailed some of my observations:
When working under the GJR-GARCH(1,1) positivity is satisfied when we impose $\omega, \beta,\alpha > 0$ and $\alpha + \gamma > 0$. The latter condition is a broader statement than imposing $\alpha, \gamma >0$, since we can allow one of the parameters to become negative (in this case, $\gamma$). I have made an in-depth answer detailing the GJR-GARCH(1,1) model, positivity, covariance stationarity and economical interpretations of the parameter estimates.
When working under the GJR-GARCH(1,1)-X we can further ensure positivity by additionally restricting the exogenous regressor such that $\omega + \nu X_{t-1} \geq 0$. Here, we allow $\nu$ to vary freely (since $\omega > 0$) as long as the above condition is satisfied. The primary motivation for the extra restriction, comes from observing the unconditional variance (calculated under assumed covariance stationarity) of the GJR-GARCH-X model:
\begin{equation}
\mathbb{V}ar(r_t) = \frac{\omega + \nu \mathbb{E}\left[X_{t-1}\right]}{1 - \alpha - \beta - \kappa \gamma}.
\end{equation}
In order to ensure non-negative unconditional variance of the return process, we specifically need $\omega + \nu \mathbb{E}\left[X_{t-1}\right] \geq 0$, which is satisfied when imposing $\omega + \nu X_{t-1} \geq 0$ for all $t$.
In conclusion, imposing $\omega, \beta, \alpha > 0$, $\alpha + \gamma > 0$ and $\omega + \nu X_{t-1}>0$, ensures that you obtain non-negative volatility estimates.
When $X_{t-1}$ is strictly positive, it is common to let $\nu \geq 0$. This is also emphasized in the article of Han, H. (2015) that investigates asymptotic results of the GARCH-X model when $X_{t-1}$ follows a fractionally integrated process. In general, it is common in academia to assume a functional form on the exogenous regressor, (see for instance (C1 - C4) in this paper p.699). This is also done in the Realized GARCH model that incorporates intraday data to procure better forecasts (I have detailed this model here, if you're interested). I hope this helps.
