Zealot barbarians don't require any material components for spells that would bring them back to life. Would this include clone as well, or just the main resurrection spells that would bring you back like revivify and raise dead?
You still need the components
The clone spell doesn't actually bring you back to life, but transfers your soul to an body identical to the one you cloned:
its soul transfers to the clone, provided that the soul is free and willing to return. The clone is physically identical to the original and has the same personality, memories, and abilities, but none of the original's equipment.
The Path of the Zealot barbarians Warrior of the Gods feature only works if a spell...
has the sole effect of restoring you to life
... which does not include clone.
Clone is cast before the character dies.
Clone spell components are still required
The description of the Clone spell says (edited for brevity; emphasis mine):
This spell grows an inert duplicate of a living, Medium creature as a safeguard against death. This clone forms inside a sealed vessel and grows to full size and maturity after 120 days; you can also choose to have the clone be a younger version of the same creature. [...]
At any time after the clone matures, if the original creature dies, its soul transfers to the clone, provided that the soul is free and willing to return.
Raise dead, and similar spells, are cast after the character dies. Clone still needs its components to be cast, since the Zealot Barbarian is alive at the time of clone's casting.
At 3rd level, your soul is marked for endless battle. If a spell, such as Raise Dead, has the sole effect of restoring you to life (but not undeath), the caster doesn’t need material Components to cast the spell on you.
That is sufficient distinction for the Zealot Barbarian's class feature not to apply to clone's material components. (Clone is more complex than "has the sole effect of restoring you to life" - the clone can (and must) exist for a period of time while the player character is alive).