How does entanglement explain myoglobin's preference for O₂?
So, there are two effects at play here which they account for. One they denote by a $\textbf{U}$ and the other by a $\textbf{J}$. $\textbf{U}$ represents many-body coulomb repulsions between the electrons. That is, they are accounting not just for two electrons repelling each other, but up to three or more. $\textbf{J}$ represents valence fluctuations in spin. Specifically, iron in hemes are in a high-spin triplet state. What they are saying then is that it is necessary to include spin fluctuations which result in changes in the magnetic moment of the system in order to properly capture the binding of ligands. If you look at figure 4 of this paper, you'll find that they have included a comparison of spin-fluctuations in $\ce{MbO2}$ and $\ce{MbCO}$ which are used as model systems and known to be analogous in ligand binding to hemes. Basically what this means is that the increased spin-fluctuations cause $\ce{O2}$ to bind more strongly, but this has basically no effect on $\ce{CO}$. They describe these effects in the following quote:


  In this case, the effect of J on the
  binding energy of O2 may be regarded as a balance between two
  competing effects. The charge analysis (Fig. 3) reveals that
  metal-to-ligand charge transfer is higher for J =0 eV, which is
  expected to enhance ligand–protein interactions for small values
  of J. However, NBO analysis reveals a larger ligand-to-metal
  back charge transfer for J = 0.7 eV, which is consistent with the
  increased occupancy of the Fe $\ce{d3z^2−r^2}$ orbital (Table 1), and is
  expected to cause variational energetic lowering at higher values
  of J due to electronic delocalization.


I won't pretend to fully understand their argument (frankly it seems to me like they are waving their hands at a solution and are basically saying this is some complicated effect having to do with spntaneously changing magnetic moments).

So, in that paper they only ascribe the effect to strong electron correlation, many-body coulomb effects, and spin-fluctuations.

So, they definitely use the word entanglement but don't define exactly what they mean or even use it as an explanatory device. I found in another one of their papers though, Weber, C., O’Regan, D. D., Hine, N. D., Littlewood, P. B., Kotliar, G., &amp; Payne, M. C. (2013). Importance of many-body effects in the kernel of hemoglobin for ligand binding. Physical review letters, 110(10), 106402., where they do actually define what they mean by entanglement and it depends on these parameters $\textbf{U}$ and $\textbf{J}$. You can find their discussion of this on page 11 in an appendix.

Their use of the word entanglement is derived from the Von Neumann Entropy which can be used as a measure of the amount of entanglement in a given system. This is because if two particles are entangled in some way, their entropy will be lower than if the two particles were not entangled. Note that this use of the word of entanglement is related to the usual descriptions of entanglement, but it is more of a measure of the average of spontaneous entanglement and disentanglement. That is, particles can become entangled due to some way in which they interact, but almost immediately interact with something else that breaks their entanglement so that it's very rare to find systems where this has any meaningful effect. This spontaneous entanglement in these heme systems is a result of the many-body effects which I've been describing. This manifests itself in the entropy they calculate. This highlights the importance of using dynamical mean-field theory (which they use) as follows:


  With this definition of the entropy, suitable for DFT
  calculations, we found that L = 0.757 in FeP-p and
  L = 0.765 in Fep-d. The entropy at the DFT level
  is hence much smaller than the DFT+DMFT entropy,
  which is in the range Λ = 2.5 − 4.2. This is explained by
  the absence of the many-body excitations induced by the
  correlations (U and J) which contribute significantly to
  the entropy.


There's no doubt they aren't being very forthcoming in their terminology or definitions, but they are also doing some very complicated and interesting stuff.