The pamphlet addresses the hare, hyrax and camel. The Torah says that kosher animals have two indicators, split hooves and chewing the cud. Rashi and other Rishonim make clear that "maaleh gairah" means to bring the food back up (ma'alah) through the neck (garon). However, the camel is a classic ruminant. The hare and hyrax do not ruminate in this manner. The apologist is left with four possible answers, none of them very satisfying.
1) The Hirsch answer: Shafan and arnevet are unknown animals that do ruminate.
2) Their mouth and nose movements appear to the lay observer like rumination.
3) The hare and hyrax do indeed process and reprocess food in a manner similar in digestion to rumination, though it is not regurgitation and cud chewing.
4) They do indeed ruminate, though we have not witnessed the matter.
Answer 1 is a punt, plain and simple. There is no basis to say this other than to dodge the contradiction. These words are used throughout Tanach and Talmud and no one questions what they are in any place except this one. Nor does Hirsch try to equate the words with any known or theoretical animal. DE has a field day with this one.
Answer 2 is more in line with what we have seen in arguing on DE before. The layman must make the determination, and it is the layman, rather than the scientist who might mistake the masticulation for rumination. This is weaker in this case, because the gemorra states, as Aish Hatorah unfortunately trumpeted in kiruv pamphlets galore, that these 4, the hare, hyrax, pig, and camel are the only four in the world with one kosher indicator but not both. Obviously, there are many animals, like kangaroos, for instance, which appear to ruminate, but do not. South American camel relatives actually do some form of rumination, but are not listed, nor would the LAYMAN refer to them as camels. This leads into what DE says in the essay, that Chazal did not have any special knowledge of animals, certainly not animals that were not discovered in their times. I would contend that when they say that these are "all the animals in the world", they are only referring to discovered animals. Obviously, they can make no conclusion about undiscovered animals. DE brings the words, "was Moshe a bowman or hunter, that he should know?" as proof that he meant all animals that would ever be discovered. However, these words emphasize the opposite. He only meant those animals that would have BEEN KNOWN by a bowman or hunter, not those that would have required prophecy. Anyway, the Torah is satisfied by this answer, but not the common interpretation of the gemorra. In fact, animals that a bowman or hunter would have known in the Middle East, rodents, appear to ruminate.
Answer 3 trivializes the matter. In the Gutnik Chumash, the editor notes that hares eat soft pellets that pass through their digestive system and come out the rectum. These are separate entities from fecal material, but technically, they are digested and redigested so the process is similar to rumination. This "cacotrophy" is unique to rabbits and hares. Similarly, the hyrax has a stomach extension that is peculiar within its kind. It could be seen as redigesting. This is sort of the opposite approach to answer two. Here we broaden the definition of maaleh gairah, so that ONLY a SCIENTIST can distinguish the indicator. IMO, this makes the indicator useless. It also trivializes the indicator. Once I broaden maaleh gairah, why am I including cacotrophy and double maw, and not including animals who eat from amongst their feces, like rats and horses. Not only that, if pigs have a split hoof, but do not chew cud, but they do eat from amongst feces, and if I extend maaleh gairah to redigestion, then why aren't pigs kosher? There is no reason why redigestion of one type should be included, but not another type. Thus, it is the same sophistry as answer 1. It answers this question, but one would not have derived the answer as a distinction without the question. Which ever animal I want to include, I just find a uniqueness about their digestion and call it maaleh gairah. It also requires the fundamentalist to go against the consensus of Rishonim in redefining maaleh gairah. All of them say that rumination means bringing it back up, as Rashi in parshas Shmini.
Answer 4 is the typical fundamentalist, Gossean burp. It gets a little support in this case, because a 1975 study claimed to confirm that hyraxes do indeed chew their cud for very short periods during a 24 hour period. The study claimed to witness the cud chewing, though I have not seen any peer review. This is still weak on grounds that it is not much of an indicator if only one study under 24 hour surveillance could show evidence, so how could the average observer ever see it. It does not help our friend, the hare, either. However, since someone says that hyraxes chew their cud, who is to say that hares do not. Maybe they do, and more studies will show that they do at some time. If the Torah says that they do, then we just don't understand how they do, but they do. Uhhhh .... OK.
R. Slifkin has already written extensively on the evidence in The Camel, The Hare and the Hyrax, so I hope this summary is helpful, but it is rehashing. So, DE has a strong point on this one. However, kiruvistim should not be hiding on these points. One, they should admit that the Chazal knew nothing of animals to which they had not been exposed and drop all proofs of Torah min HaShamayim from the 4 animals and animal based prophecies as proof. Two, they should declare that, while we do not understand how this matter correlates with Torah, we must continue to ask the question. Ultimately, with some data that does not exist today, we may obtain an answer that does indeed correlate clearly, without stretching or forcing, but we do not yet have that answer. Nor should there be a time limit, since scientific discovery is not limited by time. (This is a bit answer 4-ish, but that is OK as long as you acknowledge the weakness of it, that it has not been shown, and that any answer is speculative.) Three, they should acknowledge that such questions broaden the base of Torah learning and interest, and that only by encouraging the question to be asked, will it ever be answered. Four, they should state they they are not afraid of questions that they cannot currently answer, because they are confident that an answer will one day present itself. In short, they should acknowledge that it is a good question, and like all good questions, it is good that we have Jews knowledgeable enough to help us extend our study of Torah.
I have not used a lot of links in this, but the matters discussed are pretty open to google search.