Since the general rule is that everyone can make Attacks of Opportunity, so can Animals unless explicitly forbidden.
There is nothing in the rules, at any sane location, that prevents animals from taking Attacks of Opportunity, such as:
- Attacks of Opportunity mentions neither "Animal" nor "Intelligence"
- Animals or Creature Type: Animal doesn't contain the word "Opportunity" anywhere
- Intelligence speaks about Animal Intelligence, but only in the sense that Animal Intelligence is Int 1 or 2. It does not mention anything about Attacks of Opportunity.
Intelligence has absolutely no bearing on Attacks of Opportunity. In fact, going one step further, even Mindless creatures like Oozes or Vermin can take Attacks of Opportunity.
About the Flank trick
Your friend might be thinking of the description of the Flank trick from Handle Animal, which includes
Flank (DC 20): [...] You can instruct an animal to attack a foe you point to and always attempt to be adjacent to (and threatening) that foe. If you or an ally is also threatening the foe, the animal attempts to flank the foe, if possible. It always takes attacks of opportunity. [...]
(emphasis mine). In the context of this trick, this sentence makes little sense to me. Taking AoOs has little to do with flanking. Also, by RAW, this forces the animal to take AoOs, where without the trick it had a choice.
I find it reasonable to assume that RAI is that the animal always provokes attacks of opportunity if needed to get into flank, which also makes the context actually make sense.