If "Spirit of the Six Samurai" and the monster it is equipped to would both be destroyed at the same time (by an effect such as "Final Destiny"), the effect of "Spirit of the Six Samurai" being destroyed instead is applied, and the monster equipped with "Spirit of the Six Samurai" remains on the field.
Previously Official Rulings
These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official, but may or may not necessarily hold true nowadays.
A Union Monster that is attached [sic] to a monster is treated as an Equip Spell Card and can be destroyed with "Heavy Storm", etc. You can use "Magic Reflector" to place a counter on an equipped Union Monster and, if the equipped monster would be destroyed in battle, you can remove the counter instead so neither the equipped monster nor the Union Monster are destroyed.
If a Union Monster turns into an Equip Spell Card, or reverts back into a monster, effects are reset. For example, if I use "Call of the Haunted" to Special Summon a Union Monster, then attach [sic] it to another monster, it is no longer affected by "Call of the Haunted" (which remains on the field meaninglessly and, if destroyed, does not destroy the Union Monster). If I use "Limiter Removal" on a Union Monster, and equip it, it is not destroyed at the end of the turn, and if I turn it back into a monster its ATK is no longer doubled.
Collected Power: If multiple Union Monsters are going to be equipped to 1 legal target monster by "Collected Power", you select 1 of them and equip it to the target monster, and the other Union Monster Cards are destroyed.