  In contrast with the 3-dimensional case (cf. \cite{RaMo}), where rotationally
symmetric totally geodesic free boundary minimal surfaces have Morse index one;
we prove in this work that the Morse index of a free boundary rotationally
symmetric totally geodesic hypersurface of the $n$-dimensional Riemannnian
Schwarzschild space with respect to variations that are tangential along the
horizon is zero, for $n\geq4$.
  Moreover, we show that there exist non-compact free boundary minimal
hypersurfaces which are not totally geodesic, $n\geq 8$, with Morse index equal
to $0$. Also, it is shown that, for $n\geq4$, there exist infinitely many
non-compact free boundary minimal hypersurfaces, which are not congruent to
each other, with infinite Morse index.
  We also study the density at infinity of a free boundary minimal hypersurface
with respect to a minimal cone constructed over a minimal hypersurface of the
unit Euclidean sphere. We obtain a lower bound for the density in terms of the
area of the boundary of the hypersurface and the area of the minimal
hypersurface in the unit sphere. This lower bound is optimal in the sense that
only minimal cones achieve it.
