Pokemon X/Y Hidden Ability Breeding w/o ditto

Since you say that you're going to evolve Eevee into Sylveon, let's look at Sylveon in particular:

Sylveon has a base Speed of 60. Unfortunately, this means that Sylveon falls well below the average base Speed of most Pokemon. This contrast is well illustrated by comparing Sylveon to Eevee's other evolutions, such as Jolteon's base speed of 130 (the fastest of Eevee's evolutions) and Umbreon's base Speed of 65. In fact, Sylveon has the lowest base Speed of all of Eevee's evolutions.

This means that if you choose Charm, you will most likely go after your opponent. Therefore, your opponent will have one turn to attack with their full power, after which Charm will lower their Attack by two stages, or 50% power. So, if Sylveon can take that initial full power hit, Charm is preferable. Unfortunately, Sylveon specializes in absorbing special-type attacks, not physical. Her base Defense, sitting at 65, is half of her incredible base 130 Special Defense.

Therefore, Sylveon won't be able to tank full-powered physical moves very easily. This is where Baby-Doll Eyes' priority comes into play. Sylveon is much more likely to survive 2/3-powered physical hit as opposed to a full-powered move.

"Global Link" is just Nintendo's term that encompasses all their online services (including the Global Link Website), which includes but isn't limited to Wonder Trade, GTS and the Pokémon Dream World. In other words, every time you read "Global Link", they simply mean 'online'.

The Bulbapedia quote says it all:

"Where possible, all Pokémon obtained from the Pokémon Global Link online, either via the Pokémon Dream World or a Global Link promotion, will always have their Hidden Ability."

So unfortunately no, trading with your friend over GTS isn't going to get you the hidden ability.

However, if someone puts up a Pokemon on the GTS already with the Hidden Ability, and you trade them for it, then you will get the Hidden Ability. This is most likely how your friend got his/her Greninja with its Hidden Ability (unless someone was very kind and put it through Wonder Trade).

A brief overview of Abilities:

Abilities in general (not just Hidden ones) are generated when the Pokemon is first encountered. this means either

• In the wild,
• Hatching from an egg (passed on by the mother or the father when breeding with Ditto), or
• When evolving, If the pre-evolved form had it's hidden ability, then the evolved form will as well.

In other words, there is no way to change a Pokemon's ability via trading. (Caveat, unless there's a Pokemon that evolves when trading, and it's evolved form has different Abilities, although I cannot think of any)

So I can catch wild Pokemon with Hidden Abilities?

Only in Horde battles, and even then only rarely. Plus the types of Pokemon that appear in horde battles are few in number, and doesn't include Froakie or any other starter line at all.

So why does everyone have Hidden Abilities online?

Everyone wants Hidden Abilities as they are generally better than the Stock abilities (although this isn't always the case). Either way, there's the stigma attached to them: Because they are 'rarer', they must be 'better'. Everyone is therefore striving to get them, and that's why you see so many of them.

So where can I get Pokemon with Hidden Abilities?

As Unionhawk states, Friend Safari is your best bet. You'll need a friend of the right type (Water for Frogadier), and then Frogadier can appear in Slot 3, possibly with it's hidden ability.

Side note: One way I've found extremely helpful to check a Pokemon's ability in battle is by using the move Worry Seed, which Budew learns. Worry Seed will change the targets Ability (in battle only) to Insomnia, but before it does, it will flash up the Pokemon's Ability.