I propose "suffering requires attachment" as a VERY well developed VERY non-obvious example.

That doesn't seem like a religious claim so much as a claim originally (if so) made in a religious context. It's an (admittedly fuzzy) empirical claim about suffering, once you define your terms.

More generally, claims made within religions seem to break down into three categories: claims about God, claims about the world, and claims about God's relation/action within the world. It seems clear that Untheists would find nothing useful in the first or last of those, and it seems improper to call claims about the world "religious." Claiming the Earth is six thousand years old is not a religious claim; it's an empirical claim that some religious people believe irrespective of evidence. Similarly, true empirical claims made by religion are hardly religious by nature; if you disproved the religion to a believer, it seems unlikely he would give up those empirical beliefs, too.

Moral statements like "It's bad to hate," are a bit more complex. If they're categorical (like, "Don't murder"), they have no truth content ; I doubt Untheists would have categorical imperatives. If they're hypothetical (don't murder, or else people will murder you/society will collapse/baby Jesus will cry), then they can be evaluated empirically, and should be accepted or rejected based on supporting evidence.

More succinctly, EY's claim seems right. I doubt any strictly religious claim would be accepted by Untheists.