I suspect that the Flank trick got put in Animal Archive to help settle arguments like these.
I do wonder how Spring Attack with an animal companion is supposed to work though. Some years ago before the AA book my Druid had a hyena companion with Spring Attack. He'd often move up and bite an enemy before moving around behind that same enemy, getting into a good potential flanking position while avoiding any AoOs from the target. My PC had plenty of Handle Animal, and we assumed, perhaps incorrectly, that he could tell the hyena which square to go to when moving.
That tactic seems incompatible with the Flank trick since that says the animal always takes AoOs whereas Spring Attack would avoid those (at least from the target of the attack). Why an animal with ranks in Acrobatics couldn't use that skill while attempting to flank is a mystery to me. Maybe it is a game balance concern? Maybe it is just poor word choice or a failure to consider that some animals might have ways to avoid the AoO? I wonder if animals with an Int of 3 or higher change the situation any since you could at least try to explain to them how you'd like them to behave in combat.
Spring Attack is actually on the list of feats available even to animal companions with Int below 3. I guess it is possible that the intent is just that the animal springs to the target, attacks, and then springs back towards where it came from, but that's never spelled out in the rules.