Non-Spell Attack is not a game term, it is plain English and self explanatory. It simply means any attack that is not generated by the active casting of a spell.
So, using an attack granted by an object that is already in play is fine. However, if the attack is generated by the casting of the spell then it is a Spell Attack. Or, if the attack specifically says that it counts as a spell, such as Arcane Zap, then it is a Spell Attack (and has to go through the steps of casting a spell before you go through the attack steps).