  We consider Einstein hypersurfaces of warped products $I\times_\omega\mathbb
Q_\epsilon^n,$ where $I\subset\mathbb R$ is an open interval and $\mathbb
Q_\epsilon^n$ is the simply connected space form of dimension $n\ge 2$ and
constant sectional curvature $\epsilon\in\{-1,0,1\}.$ We show that, for all
$c\in\mathbb R$ (resp. $c>0$), there exist rotational hypersurfaces of constant
sectional curvature $c$ in $I\times_\omega\mathbb H^n$ and
$I\times_\omega\mathbb R^n$ (resp. $I\times_\omega\mathbb S^n$), provided that
$\omega$ is nonconstant. We also show that the gradient $T$ of the height
function of any Einstein hypersurface of $I\times_\omega\mathbb Q_\epsilon^n$
(if nonzero) is one of its principal directions. Then, we consider a particular
type of Einstein hypersurface of $I\times_\omega\mathbb Q_\epsilon^n$ with non
vanishing $T$ -- which we call ideal -- and prove that such a hypersurface
$\Sigma$ has either precisely two or precisely three distinct principal
curvatures everywhere. We show that, in the latter case, there exist such a
$\Sigma$ for certain warping functions $\omega,$ whereas in the former case,
$\Sigma$ is necessarily of constant sectional curvature and rotational,
regardless the warping function $\omega.$ We also characterize ideal Einstein
hypersurfaces of $I\times_\omega\mathbb Q_\epsilon^n$ with no vanishing angle
function as local graphs on families of isoparametric hypersurfaces of $\mathbb
Q_\epsilon^n.$
