  We consider $D$-dimensional amplitudes in $R^2$ gravities (conformal gravity
in $D=4$) and in the recently introduced $(DF)^2$ gauge theory, from the
perspective of the CHY formulae and ambitwistor string theory. These theories
are related through the BCJ double-copy construction, and the $(DF)^2$ gauge
theory obeys color-kinematics duality. We work out the worldsheet details of
these theories and show that they admit a formulation as integrals on the
support of the scattering equations, or alternatively, as ambitwistor string
theories. For gravity, this generalizes the work done by Berkovits and Witten
on conformal gravity to $D$ dimensions. The ambitwistor is also interpreted as
a $D$-dimensional generalization of Witten's twistor string (SYM + conformal
supergravity). As part of our ambitwistor investigation, we discover another
$(DF)^2$ gauge theory containing a photon that couples to Einstein gravity.
This theory can provide an alternative KLT description of Einstein gravity
compared to the usual Yang-Mills squared.
