If I understand this, the OP hasn't actually seen the neighbor's birds eat their eggs, it's just the neighbor claiming that they do. If they are, it's the lack of protein that usually causes it, I've had a long-time bird/chicken expert tell me that. This rings true, because it's protein that is so crucial in molting--they can't lay (at least as well) and molt simultaneously, because they need the protein for egg laying.
So, the neighbor is probably feeding scratch because it's cheap, not even giving them an "expensive" 16% layer ration, much less a decent amount of protein, so they're starved for protein.
I admit to having several integration failures with hatchery birds, but I'd try to integrate. If they've already been exposed to each other, you might not see a reason for quarantine, but the idea of getting a cheap used prefab for the new flock and setting them up a ways away from your 8x8 shed/coop and run, sounds great.
Try to "see into the future" and think about how you'd feel either way. You can't save all the neglected birds, but you might be up to trying to help the neighbor's flock. IF so, best of luck, otherwise, please don't feel bad, sounds like you have a great setup for your birds and they're paying you back with eggs and entertainment and interaction.