Permanent spells that were countered earlier in the turn never entered the battlefield, so they won't be legal targets for Brought Back.

A permanent card is an artifact, creature, enchantment, land, or planeswalker card.

If you return an Aura this way, you choose what the Aura will enchant just before it enters, but you can't choose any permanent cards that will enter at the same time as that Aura. An Aura returning to the battlefield this way doesn't target the player or permanent it will enchant, so permanents or players with hexproof may be chosen; however, the chosen recipient must be able to legally be enchanted by the Aura, so a player or permanent with protection from one of the Aura's qualities can't be chosen this way. If there's nothing legal for the Aura to enchant, it stays in the graveyard.