Gen 3 doesn’t have any shiny hunting methods other than full odds, so no, turning off your game has no effect on your chances.
The only Gen 3 game where this is somewhat untrue is Emerald, which has a glitched RNG, but even in that game, resetting your system doesn’t have too much effect unless you’re doing a SR hunt (one encounter per reset, like legendaries). If you’re interested, you can read more about Emerald’s RNG problems here. However, these problems don’t apply to FRLG, and only apply to Ruby/Sapphire if they have a dry battery, so you don’t need to worry if you’re hunting in those games.
