No, Barry's starter cannot be shiny.
It is possible for all of the following NPC Pokémon to be Shiny:
The Poochyena/Zigzagoon that attacks Professor Birch in Pokémon Ruby, Sapphire, and Emerald.
The Zigzagoon that Wally uses in the capture tutorial in Pokémon Ruby, Sapphire, and Emerald.
The Ralts that Wally catches in Pokémon Ruby, Sapphire, and Emerald (but only when he catches it and not in subsequent battles).
The Weedle that the old man encounters in Pokémon FireRed and LeafGreen.
Rental Pokémon found in the Battle Factory in Pokémon Emerald.
Wild Pokémon that appear in the Battle Pike and Battle Pyramid in Pokémon Emerald.
The Starly that attacks the player in Pokémon Diamond and Pearl, as well as in Pokémon Brilliant Diamond and Shining Pearl.
The Bidoof that Lucas or Dawn captures in the capture tutorial in Pokémon Brilliant Diamond and Shining Pearl.
Battle Frontier Pokémon used by the AI in Pokémon HeartGold, SoulSilver, and Platinum.
Professor Juniper's Minccino that appears in the "world of Pokémon" intro and the one used in the catching tutorial in Pokémon Black and White.
Professor Juniper's Cinccino that appears in the "world of Pokémon" intro in Pokémon Black 2 and White 2
The Purrloin that Bianca catches in Pokémon Black 2 and White 2.
The Fletchling that the rival, Calem or Serena, uses in the capture tutorial in Pokémon X and Y.
The Bunnelby that the rival, Calem or Serena, captures in the tutorial in Pokémon X and Y.
The only starter mentioned is Wally's, but it's only shiny for the encounter.
Source: Playing the game, shiny hunting, knowledge, watching people shiny hunt starters in Youtube videos, the comments, being smart, common sense, common knowledge, and (last but certainly not least,) this.