Decompostition of a Lipschitz domain

You may like to check such results, in particular, Proposition 2.5.4 of the below monograph. Hope you have access to it.

Proposition 2.5.4. Let $\Omega \in \mathcal A_0$ have Lipschitz boundary. Then there exists a finite open covering $\{\Omega_j\}_{j\in\{1,\dots,m\}}$ of $\overline\Omega$ such that, for every $j\in\{1,\dots,m\}$, $\Omega_j \cap \Omega$ is strongly star shaped with Lipschitz boundary.

Carbone L. and De Arcangelis R., Unbounded functionals in the calculus of variations: Representation, relaxation, and homogenization, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 125, Chapman & Hall/CRC, Boca Raton, FL, 2002

A Lipschitz domain $\Omega$ is an open set and any open set is a union of balls, which are strongly star-shaped. So I assume you meant $\overline\Omega$. By definition, any point of $\partial\Omega$ has a nbd in $\overline\Omega$ which is isometric to a sub-graph of a positive $k$-Lipschitz function $f:B(0,r)\to(0,+\infty)$, $$\{(x,t)\, :\, |x|<r, \;0<t<f(x)\}.$$ We can take $r<{f(0)\over 2k+1}$, which makes the latter set strongly star-shaped w.r.to the point $(0,r)$ as it is easy to check.