Because the animal companion is going to part of the PC's group, I would definitely use the HD/healing rules for it as well.
For monsters it's ok to completely ignore the healing rules, since 99% of the time a defeated monster is killed, and even when it's not then it's probably not coming back before some time.
But when animal companions, familiars, hirelings, allied NPCs, horses etc. are there all the time, IMO it's better to have them recover from damage, and using the same rules as the PCs is a simple starting point.