  For $m$ an even positive integer and $p$ a prime, we show that the
generalized Euler polynomial $E_{mp}^{(mp)}(x)$ is in Eisenstein form with
respect to $p$ if and only if $p$ does not divide $m (2^m-1)B_m$. As a
consequence, we deduce that at least $1/3$ of the generalized Euler polynomials
$E_n^{(n)}(x)$ are in Eisenstein form with respect to a prime $p$ dividing $n$
and, hence, irreducible over $\mathbb Q$.
