Eagles yes, spiders maybe.
Tolkien considered the possibility of mere 'beasts' possessing articulate speech - an essay in Morgoth's Ring says the Orcs were such:
In summary: I think it must be assumed that 'talking' is not necessarily the sign of the possession of a 'rational soul' or fea. The Orcs were beasts of humanized shape (to mock Men and Elves) deliberately perverted / converted into a more close resemblance to Men.
Though this essay isn't really compatible with the published 'canon' it does show that Tolkien considered the possibility of nonrational creatures speaking, and also that fea is equivalent to the concept of 'rational soul' in our-world theology/philosophy.
Also from a letter (no. 153, but quoted in Morgoth's Ring) Tolkien said
I am not sure about Trolls. I think they are mere 'counterfeits'
Of course... when you make Trolls speak you are giving them a power, which in our world (probably) connotes the possession of a 'soul'.
Therefore we can say that anything with a fea is rational.
The Great Eagles are "spirits from afar" inhabiting animal bodies, sent (along with the spirits that became Ents) apparently by Eru as a result of Yavanna's discussion with Manwe about protecting nature. [This is described in "Of Aule and Yavanna" ch. 2 of the Silmarillion.] They're probably closer in nature to Gandalf or Saruman than normal birds.
The Spiders of Mirkwood are not natural spiders, and not just in size: they are descendants of Shelob:
her lesser broods, bastards of the miserable mates, her own offspring, that she slew, spread from glen to glen, from the Ephel Dúath to the eastern hills, to Dol Guldur and the fastnesses of Mirkwood.
And as Shelob is the "last child of Ungoliant to trouble the unhappy world", and Ungoliant was probably one of the Ainur and definitely a very powerful evil spirit, there is a 'supernatural' or 'spiritual' strain in the spiders of Mirkwood.
Whether they're completely rational in the sense that humans (and Elves, Dwarves etc.) are is probably unanswerable -- Tolkien's notes on this sort of thing in History of Middle-Earth vol. X - Morgoth's Ring are inconclusive/somewhat undecided as to whether speaking actually implies possession of a fea or rational soul -- but smarter than ordinary spiders, for sure.