Probably not. The spellslinger archetype says it lets you cast any of your spells through your gun. An SLA is not a spell, it just behaves like a spell. Since metamagic rods don't work with SLAs, I'd say this also does not work with SLAs.
Spell-like abilities are not actually spells, hence the name. The Arcane Gun archetype feature you're looking at is only concerned with actual spells being cast through the gun. So, no, doesn't apply.