Age old question.

Does Gary oaks Pokemon go into it's pokeball? The theory is that ashes Pikachu does not go into a pokeball because none of the pokeballs have the lightning symbol that was on the first pokeball ash got Pikachu in. And as you would notice in the first episode all of the other pokeballs are still with the professor. Does this mean that Gary cannot put Squirtle in it's pokeball or is this theory pretty much debunked?

Ah, he was shown using his Pokeball to send out Blastoise, so it's pretty clear Gary did in fact keep his Squirtle in a Pokeball.