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 Renyi entanglement entropy $S_2$. Our results show a cross-over from volume scaling, $S_2sim ell^theta$ ($theta=2$ in $d=2$ dimensions), to the characteristic Fermi-liquid behavior $S_2sim ellln ell$ 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.
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