Can an animal companion, without a handle animal check, carry your gear like a mule would?
Is there a minimum required Int score for this action, in that case?
Yes, as camel, light, and heavy warhorses are listed as animal companions.
It is reasonable to assume that certain animal companions "completely typical for its kind" (PHB) such as riding dogs would be able/be trained/know how to haul equipment. More complex are cases of non-working animals such as wolves.
An animal companion could be trained to fulfill roles or preform tricks (using the handle animal check) and then later simply ordered to do so. Wolves do not typically haul equipment and must be trained or pushed with handle animal checks to haul equipment or even fetch, guard, or stay. Consult DM for campaign or house standards.
No, and mules don't either. The question is mistaken, in that a mule still requires handle animal checks to carry gear. There is no alternate mechanism described.
Handle Animal You can train a mule (or other animal) any way you feel like, but the generic mule is probably trained for the "Heavy Labor" purpose. (d20SRD: Handle Animal), and so knows the tricks "come" and "work." (Because that purpose is easiest and fastest, and useful enough for merchants to sell). In general, it is DC 10 to make an animal perform a trick which it knows, or raw charisma if Handle Animal is untrained (ibid). Anyone without a charisma penalty can take ten in to make an animal perform a trick it knows in normal circumstances, and anyone with at least a +2 can take ten in the majority of circumstances (because the DC is rarely more than 12). However, with a net -1 or greater, nothing suggests that a character can routinely get an animal (including a mule) to perform tricks.
Animal Companions Animal companions work like any other animal, but they also get a few relevant bonuses: - Extra tricks, per the animal companion table. Normally an animal gets three tricks at 1 intelligence and six at 2, but animal companions get extras. - The owner gets a +4 bonus to Handle Animal checks with that animal. - The owner can push the animal (make it perform a trick which it doesn't know) , even if handle animal is untrained. - Faster handling: normally handling is a move action and pushing is full-round. for an animal companion, handling is free and pushing is a move action.
Barding There's one last important caveat: if an animal wears barding, it "cannot be used to carry any load other than the rider and normal saddlebags." (d20SRD: Barding) You almost certainly want your animal companion in barding, so that may affect the practical application of this answer.
So How Do I Make It Carry My Stuff? Depending how you trained your animal companion, you either take free actions to make him do tricks he knows or take move actions to get him to do tricks he doesn't know. That's... pretty much it.
If your DM thinks that the animal would try to get rid of the load (reasonable, but not mechanically supported), I would argue that the worst case is a single DC 25 handle animal check (move action) to push the animal into doing the "work" trick. There is nothing preventing you from taking ten or twenty on this check.