Most likely.
All animal companions (I think?) have a neck slot, which can be used to wear an amulet of mighty fists.
The questionable part comes in the description of the flaming quality: "Upon command, a flaming weapon is sheathed in fire that deals an extra 1d6 points of fire damage on a successful hit. The fire does not harm the wielder. The effect remains until another command is given."
Animals can't really issue command words.
If the flaming quality remains active on the amulet of mighty fists, even when handed around, then you could activate it yourself and then hand it over to the animal. If it 'resets' when moved between creatures, or if it specifically activates the creature rather than itself, then it wouldn't work.
An animal companion which gains the ability to speak could activate such an amulet via command word. I'm not sure if animal companions who can't speak could use it though. I suppose they might be able to howl and try the UMD skill, but that's not a skill most animal companions are very good at (though dinosaurs have enough Charisma to be decent and the idea of a T-Rex who uses wands with his tiny arms seems amusing to me)