Let's call it: Eudoxus’ argument from universal pursuit of pleasure (1172b9–15):
Εὔδοξος μὲν οὖν τὴν ἡδονὴν τἀγαθὸν ᾤετ' εἶναι διὰ τὸ πάνθ' ὁρᾶν ἐφιέμενα αὐτῆς, καὶ ἔλλογα καὶ ἄλογα, ἐν πᾶσι δ' εἶναι τὸ αἱρετὸν τὸ ἐπιεικές, καὶ τὸ μάλιστα κράτιστον· τὸ δὴ πάντ' ἐπὶ ταὐτὸ φέρεσθαι μηνύειν ὡς πᾶσι τοῦτο ἄριστον ὄν· ἕκαστον γὰρ τὸ αὑτῷ ἀγαθὸν εὑρίσκειν, ὥσπερ καὶ τροφήν, τὸ δὲ πᾶσιν ἀγαθόν, καὶ οὗ πάντ' ἐφίεται, τἀγαθὸν εἶναι. 
Eudoxus thought pleasure the good because of seeing all animals aim at it, both rational and non-rational, and because what is choiceworthy in all cases is what is fitting and what is particularly choiceworthy is most powerful. The fact that they all are attracted to the same object suggests that this is best for all things. For each finds what is good for it, as it also does food, but that at which all things aim it is the good.
My first question is: What is Eudoxus’ argument? Eudoxus combines an observation that all creatures, both rational and non-rational, pursue pleasure with an argument to the effect that what all creatures pursue is the good. He begins with a general premise about that all animals are pursuers of what is good for them. This is a general thesis about animal motivation, supported with a further observation. Each species pursues its natural good, something that is obvious when we consider as an example the way in which each species pursues its own particular diet. Squirrels look for nuts; lions hunt antelopes. This might suggest an argument along the following lines: All animals pursue radically different ends. But in fact, there is one thing which all animals – rational and non-rational – pursue, namely pleasure. This is as true of squirrels and lions as it is of humans. In this way Eudoxus is hoping to move beyond a mere descriptive claim about what all animals do in fact pursue to a normative claim about what is good for all animals and therefore good for humans too.
However, there is a problem. If we can rely with some confidence on Aristotle’s presentation of the argument, this is not quite what Eudoxus says. He does not opt for a starting descriptive premise of the sort familiar from some later Hellenistic philosophers. He does not, for example, claim like the Epicureans that all animals aim solely and instinctively at pleasure (perhaps some form of psychological hedonism or the view that pleasure is the natural goal of all unreflective pursuit). Nor, in Aristotle’s version of the argument, does he offer the explicit claim that although animals may aim at a variety of different things depending on circumstances, pleasure is the only thing at which all animals aim (although perhaps this might be the implicit thought behind the very last clause in the section just cited). That latter claim is, however, a reasonable interpretation of the argument ascribed to Eudoxus in the report by Alexander of Aphrodisias (In Arist. Top. p.226.16–18 Wallies):
Εὔδοξος ἐδείκνυε τὴν ἡδονὴν τὸ μέγιστον τῶν ἀγαθῶν ἀπὸ τοῦ πάντα μὲν τὰ ζῷα ταύτην αἱρεῖσθαι, μηδὲν δὲ τῶν ἄλλων ἀγαθῶν κοινὴν οὕτως ἔχειν τὴν αἵρεσιν.
Eudoxus demonstrated that pleasure was the greatest good from the fact that all animals choose it and that no other good is chosen so generally.
Aristotle himself makes no comment about whether pleasure alone or pleasure especially is chosen by a wide range of animals, let alone whether it alone is chosen by all animals.  We have no reason to suppose that Alexander had access to Eudoxus’ works or philosophy beyond what he could find in Aristotle, so there is no reason to prefer his later presentation to that given in EN X.2. But it is not difficult to see why Eudoxus might be thought by Alexander to have argued along those lines, and therefore why his account of the argument, although based on Aristotle, is subtly but significantly different. Had Eudoxus offered either of these stronger claims, namely (i) that pleasure is the only good at which animals aim or (ii) that pleasure is the only good at which all animals aim, then he might have been able more easily to go on to conclude that, since all animals desire what is good for themselves, and the one thing which all animals desire is pleasure, then we have good reason to conclude that pleasure is good for all animals and indeed that it is the only thing that is good for all animals. In that case, pleasure is the good for each and every animal qua animal. In the absence of the stronger claim of the uniquely universal pursuit of pleasure, Eudoxus would have to reach for something else to move from his descriptive premise to his desired conclusion.
Rather, according to Aristotle’s interpretation, Eudoxus’ argument seems to begin merely with the claim that pleasure is an object of pursuit shared by all animals. The important argumentative work, in that case, must be done by the curious intermediate premise, generally not emphasised by the later commentators’ versions of the argument, that ‘what is choiceworthy in all cases is what is fitting and what is particularly choiceworthy is most powerful’. Unfortunately, the precise meaning – and indeed, due to Aristotle’s characteristic concision, the correct translation – of this inference is itself unclear. That's a worry for another time.
For now, I'm interested in Aristotle's reaction at 1175b35-1173a5.
Aristotle is evidently trying to perform some kind of a salvage operation on the Eudoxan argument, and he does so principally by wondering whether it points to a natural and shared tendency among all animals, rational or otherwise. Eudoxus’ insight is to stress how pleasure is sought not only by non-rational animals, but by rational animals too. Given this additional class of pleasure-seekers, it becomes impossible to conclude that pleasure-seeking is a mere brutish activity unsuitable for rarefied creatures such as ourselves. Of course, there are some distinctions to be drawn between the pursuit of pleasure by, say, a cat and the pursuit of pleasure by an Athenian aristocrat, but Aristotle is prepared to speculate that the cat’s aim for the pleasures of a place by the fire may be an indication of something rather interesting. Perhaps, he wonders, even in the lower, non-rational creatures there is some natural good which aims them at the good appropriate for them. Presumably, he means something like the following: These lower creatures cannot reason about what is their own proper (οἰκείον) good; but pleasure may well serve as a mechanism for encouraging or driving them nevertheless to pursue what they ought. My cat, for example, cannot deliberate about what is good for it nor can it engage in any sophisticated deliberation about whether it should sit by the fire on a rainy night. Nevertheless, the fact that it takes pleasure in warmth and comfort means it pursues a good which is proper to it qua cat.  In effect, Aristotle is exploring the possibility that we can make sense of Eudoxus’ argument not as an attempt to belittle the behaviour of rational animals by stressing something they have in common with their non-rational fellows, but as an indication that non-rational animals too may have a natural tendency to orient themselves towards what it good. The fact that pleasure attracts both the rational and the non-rational gives us a good reason for thinking that it is a good, if not the good. 
 Bywater brackets ἕκαστον γὰρ τὸ αὑτῷ ἀγαθὸν εὑρίσκειν, ὥσπερ καὶ τροφήν, but this seems to me to be part of an inference that includes the whole remainder of the cited passage.
 Cf. Heliodorus Paraphr. Eth. Nic. p.210.25–7 Heylbut: πᾶσι δὲ κοινῶς ἀγαθὸν οὗ πάντα κοινῶς ἐφίενται καὶ πορίζειν βούλονται ἑαυτοῖς· ὃ δὲ πᾶσίν ἐστιν ἁπλῶς ἀγαθὸν καὶ οὗ πάντα ἐφίενται, τοῦτο εἶναι τὸ ἔσχατον ἀγαθόν.
 Michael of Ephesus is prepared to make a more extravagant teleological claim: ἔνεστι γὰρ ἐν ἅπασιν ἢ νοῦς ἢ νοῦ τις αὐγὴ καὶ ἔλλαμψις, ὡς αὐτὸς ἐν ἄλλοις ἔδειξε, καὶ “φύσιες ζῴων ἀδίδακτοι”, ὡς Ἱπποκράτης εἴρηκεν (In Arist. Nic. Eth. 534.15–17 Heylbut).
 Cf. Heliodorus Paraphr. Eth. Nic. p.211.28–36 Heylbut.