  We construct embedded ancient solutions to mean curvature flow related to
certain classes of unstable minimal hypersurfaces in $\mathbb{R}^{n+1}$ for $n
\geq 2$. These provide examples of mean convex yet nonconvex ancient solutions
that are not solitons, meaning that they do not evolve by rigid motions or
homotheties. Moreover, we construct embedded eternal solutions to mean
curvature flow in $\mathbb{R}^{n+1}$ for $n \geq 2$. These eternal solutions
are not solitons, are $O(n)\times O(1)$-invariant, and are mean convex yet
nonconvex. They flow out of the catenoid and are the rotation of a profile
curve which becomes infinitely far from the axis of rotation. As $t \to
\infty$, the profile curves converge to a grim reaper for $n \geq 3$ and become
flat for $n=2$. Concerning these eternal solutions, we also show they are
asymptotically unique up to scale among the embedded $O(n)\times
O(1)$-invariant, eternal solutions with uniformly bounded curvature and a sign
on mean curvature.
