  In this work we study the asymptotically flat, static, and spherically
symmetric black-hole solutions of the theory described by the action $$S = \int
d^nx\sqrt{-g} \left\{\left(1-\xi\phi^2 \right)R -
g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi\right\},$$ with $n>3$ and arbitrary
$\xi$. We demonstrate the absence of scalar hairs for $\xi<0$. For
$\xi>\xi_c=\frac{n-2}{4(n-1)}$, we show that there is no scalar hair obeying
$|\phi(r)| < 1/\sqrt{\xi}$ or $|\phi(r)| > 1/\sqrt{\xi}$. For $0<\xi<\xi_c$, we
prove the absence of scalar hairs such that $|\phi(r)| < 1/\sqrt{\xi}$ or
$\frac{1}{\xi} < \phi^2(r) < \frac{\xi_c}{\xi(\xi_c-\xi)}$.
