Disclaimer: more a long comment than an answer. I can only say the Szegő kernel $S$ exists even if $\Omega$ is "only" a piecewise smooth bounded domain (no need for pseudoconvexity), as an old result by Lutz Bungart ([2] §5 p. 1157-1158, theorem 5.1) also found in Aizenberg and Yuzhakov's monograph ([1] chapter 1, §6 p.48, theorem 6.7) (this reference is possibly the most complete reference on Szegő kernel I'm aware of): the result is precisely the following

**Theorem**. Let $f\in A_C(\Omega)$ where $\Omega$ is a piecewise smooth bounded domain in $\Bbb C^n$, $n>1$. Then there exist an integral representation
$$
f(w) =\int_{\partial \Omega} f(\bar\zeta) S(\bar\zeta, w) \operatorname{d\!}\sigma_\zeta
$$
whose kernel $S(\bar\zeta, w)\in \mathscr{H}\big(\{(\bar\zeta, w): w\in\Omega,\;\zeta\in\Omega\}\big)$ is in $H^2(\operatorname{d\!}\sigma_\zeta)\equiv H^2(\partial\Omega)$ (here we have used $\mathscr{H} (D)$ tho denote the vector space of holomorphic function on an open domain $D$).

Thus, in general, the only thing you can expect is that $S(\cdot,w)$ has a radial limit a.e. on $\partial\Omega$, and not that it extends continuously from $\Omega$ to its boundary: for example, a(n almost) classical result ([3] chapter II, §10 pp. 38-40 theorem 10) says that this happens if $\Omega\in C^2$.

**References**

[1] L. A. Aizenberg and A. P. 3, *Integral representations and residues in multidimensional complex analysis*, translated from the Russian by H. H. , ed. by Lev J. Leifman, (English), Translations of Mathematical Monographs, 58. Providence, R.I.: American Mathematical Society (AMS), pp. X+283 (1983), ISBN: 0-8218-4511-X, MR0735793, Zbl 0537.32002.

[2] Lutz Bungart, "Boundary kernel functions for domains on complex manifolds" (English), Pacific Journal of Mathematics 14, 1151-1164 (1964), MR0174783, Zbl 0144.08001.

[3] Elias M. Stein, *Boundary behavior of holomorphic functions of several complex variables* (English), Mathematical Notes, Princeton, N. J.: Princeton University Press, pp. IX+72 (1972), MR0473215, Zbl 0242.32005.