A counterexample: $n=2$, $$\Omega=B^-_{(0,0)}(2)\cup B^+_{(1,0)}(1)\cup B^+_{(-1,0)}(1),$$ where $B^\pm_C(r):=B_C(r)\cap\Pi^\pm$, $B_C(r)$ is the open ball of radius $r$ centered at $C$, and $\Pi^\pm:=\{(x,y)\in\Bbb R^2\colon\pm y\ge0\}$.
Here is a picture of $\Omega$, with the bad point $(0,0)$ on its boundary:
A counterexample: $n=2$, $$\Omega=B^-_{(0,0)}(2)\cup B^+_{(1,0)}(1)\cup B^+_{(-1,0)}(1),$$ where $B^\pm_C(r):=B_C(r)\cap\Pi^\pm$, $B_C(r)$ is the open ball of radius $r$ centered at $C$, and $\Pi^\pm:=\{(x,y)\in\Bbb R^2\colon\pm y\ge0\}$.
A counterexample: $n=2$, $$\Omega=B^-_{(0,0)}(2)\cup B^+_{(1,0)}(1)\cup B^+_{(-1,0)}(1),$$ where $B^\pm_C(r):=B_C(r)\cap\Pi^\pm$, $B_C(r)$ is the open ball of radius $r$ centered at $C$, and $\Pi^\pm:=\{(x,y)\in\Bbb R^2\colon\pm y\ge0\}$.
Here is a picture of $\Omega$, with the bad point $(0,0)$ on its boundary:
