Does the animal companion's Share Spells feature allow a druid/ranger who has fast healing to share it with their animal companion?
No, fast healing is not shared with animal companions.
From the PFSRD, on Share Spells (emphasis mine):
The druid may cast a spell with a target of “You” on her animal companion (as a touch range spell) instead of on herself. A druid may cast spells on her animal companion even if the spells normally do not affect creatures of the companion’s type (animal). Spells cast in this way must come from a class that grants an animal companion. This ability does not allow the animal to share abilities that are not spells, even if they function like spells.
Share Spells only works for spells (not spell-like abilities or anything else) that 1) have a range of "You"/"Self"/"Personal" (i.e. spells that can normally only target the caster), and 2) come from a class that grants an animal companion.
If you're getting fast healing from a self-only druid/hunter/ranger spell, then you can also cast that spell on your companion. If you have fast healing from any other source then you cannot give it to or share it with your animal companion via Share Spells.