Can Pokemon from the Game Corner have their Hidden Ability, or is that something you have to breed for?
The pokemon from the Game Corner only have their regular abilities.
If you want pokemon with hidden abilities:
Go to a friend safari that has the mons you are looking for, the mons from friend safari have a 1/5 chance of having their hidden ability. The mons you want are unlocked after you get the sixt gym badge.
In the post game go to Gaea Town and give a Dream Mist to the Dream Guru to change the ability of a Pokemon for their Hidden one.