What happens if I use Steelix's "Gigaton Shake" attack, and then my opponent uses Mawile GX's "Captivating Wink" Ability during their turn? Would those Pokemon I put into play be able to attack on my next turn (since it was not during MY turn)?
No, they would not be able to attack.
Source: Unified Minds FAQ (2019-07-19), TPCi Rules Team (2019-08-01)

Relevant cards

Note: All applicable cards may not be listed below.
SM11_104
SM11_104
SM11_141
SM11_141
SM11_227
SM11_227
SM11_246
SM11_246