  Let $F \in \mathbb{Z}[x,z]$ be a binary form of degree $2n+1 \geq 5$. A
result of Darmon and Granville known as "Faltings plus epsilon" states that
when $F$ is separable, the superelliptic equation $y^2 = F(x,z)$ has finitely
many primitive integer solutions. In this paper, we prove a strong asymptotic
version of "Faltings plus epsilon" which states that in families of
superelliptic equations of sufficiently large degree and having a fixed
non-square leading coefficient, a positive proportion of members have no
primitive integer solutions (put another way, a positive proportion of the
corresponding binary forms fail to properly represent a square). Moreover, we
show that in these families, a positive proportion of everywhere locally
soluble members have a Brauer-Manin obstruction to satisfying the Hasse
principle. Our result can be viewed as an analogue for superelliptic equations
of Bhargava's result that most even-degree hyperelliptic curves over
$\mathbb{Q}$ have no rational points.
