  Cais, Ellenberg and Zureick-Brown recently observed that over finite fields
of characteristic two, all sufficiently general smooth plane projective curves
of a given odd degree admit a non-trivial rational 2-torsion point on their
Jacobian. We extend their observation to curves given by Laurent polynomials
with a fixed Newton polygon, provided that the polygon satisfies a certain
combinatorial property. We also show that in each of these cases, the
sufficiently general condition is implied by being ordinary. Our treatment
includes many classical families, such as hyperelliptic curves of odd genus and
$C_{a,b}$ curves. In the hyperelliptic case, we provide alternative proofs
using an explicit description of the 2-torsion subgroup.
