  Segre surfaces in the title mean quartic surfaces in $\mathbb{CP}^4$ which
are the images of weak del Pezzo surfaces of degree four under the
anti-canonical map. We first show that minimal minitwistor spaces with genus
one are exactly Segre quartic surfaces. By a kind of Penrose correspondence,
Zariski open subsets of the projective dual varieties of these surfaces admit
Einstein-Weyl structure. We investigate structures of these dual varieties in
detail. In particular, we determine the degrees of these varieties (namely the
classes of the Segre surfaces), as well as structure of several components of
the boundary divisors which are the complements of the Einstein-Weyl spaces in
the projective dual varieties.
