Given a point on the boundary of $\Omega$, choose an orthogonal coordinate system $x_1,\ldots,x_n$ for $\mathbb{R}^n$ so that the point lies at $(0,0,\ldots,0)$ and the tangent hyperplane to $\partial\Omega$ at the point is the hyperplane $x_n = 0$.
In this case, by the Implicit Function Theorem, there is an $\epsilon$-neighborhood of the origin on which $\Omega$ is defined by $ x_n \;<\; f(x_1,\ldots,x_{n-1}) $ where $f\colon\mathbb{R}^{n-1}\to\mathbb{R}$ is a $C^2$ function and $f(0,\ldots,0)=0$. This function has a Hessian matrix $Hf$ at the origin, where $ (Hf)_{ij} \;=\; \frac{\partial f}{\partial x_i\partial x_j}(0,\ldots,0). $ This matrix is symmetric, and hence diagonalizable, with real eigenvalues $\lambda_1,\ldots,\lambda_n$. (These are the principal curvatures of $\partial\Omega$ at the origin.) Choose any radius $0 satisfying $ \frac{1}{r} \;\geq\; \max\{\lambda_1,\ldots,\lambda_n\}. $ Then the closed ball of radius $r$ centered at the point $(0,\ldots,0,r)$ will intersect $\overline{\Omega}$ only at the origin.