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Here a domain is connected. Suppose both a negative point and a zero point exist. The open set $D=\{v<0\}$ then has a boundary point $q$ inside $\Omega$. Choose $z\in D$ close enough to $q$ that the Euclidean ball of radius $R=\mathrm{dist}(z,Z)$ stays compactly inside $\Omega$; it is contained in $D$ and touches $Z$ at some $y$. Attainment of this distance follows by restricting to a compact neighbourhood of $q$.

On that Euclidean ball $v$ is twice differentiable andThe Hopf boundary point lemma gives a strictly positive outward radial derivative at $y$. But $y$ is an interior maximum of the $C^1$ function $v$ on $\Omega$, so $Dv(y)=0$. This contradiction proves the Hopf dichotomy with classical regularity away from the zero set:

For the requested weaker-regularity counterexample take $\Omega =(-1,1{)}^n$ andIt is a locally Lipschitz function and nonpositive. Its negative set is $x_1>0$, where $v=-\sin x_1$ is smooth and $\Delta v+v=0$. On $x_1\le 0$ it is zero. The normal derivative jumps from zero to minus one across $x_1=0$, so it is not $C^1$. All other hypotheses hold; imposing the equation also across that interface would be an additional condition absent from the question.