He really has nothing new to say. He provides no evidence for his claims, just a series of assertions:
"Language in animals has never been demonstrated": actually, it has. I gave several citations, but Egnor didn't address any of them. There is a whole subfield of ethology that deals with this. Egnor didn't even seem to know the name of the field.
"because animals are incapable of language.": pure assertion.
"Claims of animal language have been made by some ethologists, but those claims are mistaken": Egnor suddenly starts using the word "ethologist", which he didn't before. I am glad to have informed him of the name for the practitioners of the field he is criticizing. Again, pure assertion. He doesn't actually address any of the cited studies.
"We should begin with an examination of what we mean by language.": Egnor is not a linguist, either.
"The confusion between signals and designators is at the root of ethologists' misunderstanding about animal "language.": Yeah, all those ethologists who actually study animal language are wrong, but Egnor (who didn't seem to even know the word "ethologist" until two days ago) is right, despite not working in the field. Remember the word "egnorance" and why it was coined?
"Natural animal signals have no grammar": Probably not true. For example, see here and here and here. Now one could certainly take issue with any or all of these, but the point is that there is a large literature that needs to be assessed carefully, and which cannot be addressed by categorical denials of the kind Egnor makes. Egnor does not do this. He just knows he is correct, because ... Aristotle.
"Animals do not signal abstract concepts": pure assertion.
(Quoting de Waal): "We honestly have no evidence for symbolic communication, equally rich and multifunctional as ours, outside our species": A red herring. Nobody said animal language was as "rich and multifunctional" as human language, just that it exists, contrary to Egnor's claims. This is the traditional creationist technique known as "moving the goalposts".
Finally, Egnor insinuates that I haven't read de Waal's books, when I was the one who introduced them to him. He urges me to read de Waal's books. I have. My records show that I read Good Natured in 1996, as well as Chimpanzee Politics and Peacemaking Among Primates.