Once an outlaw that came under your control this turn has legally attacked, causing it to lose haste by removing Hellspur Posse Boss won’t cause that outlaw to stop attacking.

A card, spell, or permanent is an outlaw if it has the Assassin, Mercenary, Pirate, Rogue, or Warlock creature type. It doesn’t matter if it has more than one of those creature types; as long as it has at least one, it’s an outlaw.

Outlaw is not a creature type. If an effect asks you to choose a creature type, you can’t choose outlaw.

If an ability refers to an outlaw or whether a player controls an outlaw, it’s referring only to permanents with one or more of the creature types specified above. Notably, it’s not referring to any spell or card not on the battlefield. However, other abilities may refer to an “outlaw spell” or “outlaw card” in a zone other than the battlefield. Those abilities refer to spells and cards with one or more of the specified creature types.