If I use Greninja's "Water Shuriken" ability first and then evolve to Greninja BREAK on the same turn, would I be able to use its "Water Shuriken" ability again?

No, you cannot. Even though Greninja evolved into Greninja BREAK and retains the Abilities of its previous evolution, it still remembers that it has already used up its once-per-turn Water Shuriken ability for that turn.

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