Something to remember is that an Amulet of Mighty Fists specifically states "This amulet grants an enhancement bonus of +1 to +5 on attack and damage rolls with unarmed attacks and natural weapons." It also lets you add weapon special properties to unarmed and natural attacks. Animals have Natural weapons, so it would stand to reason than any animal would be able to use such an item. The problem with animals and some items is that they cannot (for obvious reasons) activate magical items, so any items they could use would have to be of the "always on when worn" variety. You could argue that an item had been specifically designed to be activated by an animal, but you would need to use one of the animal's tricks to teach it how to use it, and then command the animal to turn it on or off.
Also note that unless your playing in a world where Druids and Wizards are constantly outfitting all of their animals with these things, finding magical equipment for an Animal is likely to be a rare thing. Players may need to commission the construction of these specialty magical items, something that requires time (finding a person who can create a masterwork item of the required type, finding the person who can enchant it, negotiating with these people, plus the time required to craft such things) and raw resources the players may or may not not have. If your Druid/Ranger wants magical items for their pets, this could easily become a quest in and of itself.
I do like the 12-slot list given by mxyzplk. I think it would give a character many options for outfitting his animal companion.