  The Euler numbers occur in the Taylor expansion of $\tan(x)+\sec(x)$. Since
Stieltjes, continued fractions and Hankel determinants of the even Euler
numbers, on the one hand, of the odd Euler numbers, on the other hand, have
been widely studied separately. However, no Hankel determinants of the (mixed)
Euler numbers have been obtained and explicitly calculated. The reason for that
is that some Hankel determinants of the Euler numbers are null. This implies
that the Jacobi continued fraction of the Euler numbers does not exist. In the
present paper, this obstacle is bypassed by using the Hankel continued
fraction, instead of the $J$-fraction. Consequently, an explicit formula for
the Hankel determinants of the Euler numbers is being derived, as well as a
full list of Hankel continued fractions and Hankel determinants involving Euler
numbers. Finally, a new $q$-analog of the Euler numbers $E_n(q)$ based on our
continued fraction is proposed. We obtain an explicit formula for $E_n(-1)$ and
prove a conjecture by R. J. Mathar on these numbers.
