  The fermion sign problem is often viewed as a sheer inconvenience that
plagues numerical studies of strongly interacting electron systems. Only
recently, it has been suggested that fermion signs are fundamental for the
universal behavior of critical metallic systems and crucially enhance their
degree of quantum entanglement. In this work we explore potential connections
between emergent scale invariance of fermion sign structures and scaling
properties of bipartite entanglement entropies. Our analysis is based on a
wavefunction ansatz that incorporates collective, long-range backflow
correlations into fermionic Slater determinants. Such wavefunctions mimic the
collapse of a Fermi liquid at a quantum critical point. Their nodal surfaces --
a representation of the fermion sign structure in many-particle configurations
space -- show fractal behavior up to a length scale $\xi$ that diverges at a
critical backflow strength. We show that the Hausdorff dimension of the fractal
nodal surface depends on $\xi$, the number of fermions and the exponent of the
backflow. For the same wavefunctions we numerically calculate the second
R\'enyi entanglement entropy $S_2$. Our results show a cross-over from volume
scaling, $S_2\sim \ell^\theta$ ($\theta=2$ in $d=2$ dimensions), to the
characteristic Fermi-liquid behavior $S_2\sim \ell\ln \ell$ on scales larger
than $\xi$. We find that volume scaling of the entanglement entropy is a robust
feature of critical backflow fermions, independent of the backflow exponent and
hence the fractal dimension of the scale invariant sign structure.
