Consider the SDE: \begin{equation} dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} \end{equation} It describes a process $v_t$ which is a strictly positive if the drift is stronger enough, i.e. if drift parameters ($k$, the speed of mean-reverting, and $\theta$, mean-reverting level) and the Vol-of-Vol $\xi$ satisfy: \begin{equation} k \theta > \frac{1}{2} \xi^2 \end{equation} which is known as Feller condition. I know this condition can be generalized to multi-factor affine processes. For example, if the volatility of the returns $\log S_t$ is made of several independent factors $v_{1,t},v_{2,t},...,v_{n,t}$, then the Feller condition applies to each factor separately (check here at page 705, for example). Moreover Duffie and Kan (1996) provide a multidimensional extension of the Feller condition.
But I still don't understand if we still need the (or a sort of) Feller condition in case of jump-diffusion. You may consider for example the simple case of a volatility factor with exponentially distributed jumps: \begin{equation} dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} + dJ^{v}_{t} \end{equation} where $J^{v}_{t}$ is a compound Poisson process, independent of the Wiener $W^{v}_{t}$. The Poisson arrival intensity is a constant $\lambda$ with mean $\gamma$. I observe that in this case, the long term mean reverting level is jump-adjusted: \begin{equation} \theta \Longrightarrow \theta ^{*}=\theta + \frac{\lambda}{k} \gamma \end{equation} so I suspect if a sort of Feller condition applies it must depends on jumps.
Nevertheless, from a purely intuitive perspective, even if the barrier at $v_t = 0$ is absorbent, jump would pull back from 0 again.
Thanks for your time and attention.
Are you looking for a condition to ensure that the origin is not attainable or one to ensure that process is nonexplosive?
If the former, note that in your jump-diffusion, the size of the jumps must be nonnegative. Otherwise, it's possible that one negative jump leads the process to achieve some negative position, which makes the square root term undefined. Therefore, with the assumption of nonnegative jumps, the same Feller condition ensures that the process never achieves the origin in finite time.
If the latter, note that the jumps cannot occur too frequently ($\lambda$ should be small enough) and that the jump size cannot be too large ($\gamma$ should be small). Otherwise, the process may blow up. More precisely, the mean-reversion must be "stronger" than the jumps for nonexplosiveness. If the intensity of the jumps is $\lambda_1 v_t + \lambda$, then the following condition $$\kappa > \lambda_1\gamma$$ would make the process not only nonexplosive, but in fact ergodic.
Since the intensity is a constant in your setting, you don't need such a condition.