  In this paper we obtain two classification theorems for free boundary minimal
hypersurfaces outside of the unit ball (exterior FBMH for short) in Euclidean
space. The first result states that the only exterior stable FBMH with parallel
embedded regular ends are the catenoidal hypersurfaces. To achieve this we
prove a B\^ocher type result for positive Jacobi functions on regular minimal
ends in $\mathbb{R}^{n+1}$ which, after some calculations, implies the first
theorem. The second theorem states that any exterior FBMH $\Sigma$ with one
regular end is a catenoidal hypersurface. Its proof is based on a
symmetrization procedure similar to R. Schoen [14]. We also give a complete
description of the catenoidal hypersurfaces, including the calculation of their
indices.
