Some of this is newer information, some of it 3-4+ days old with no SS evidence to prove it wrong.

Certain mob jobs are staggered by ws (Pld Sam Drk Drg War)
Certain mob jobs are staggered by ja's (Thf Nin Rng Mnk Bst)
Certain mob jobs are staggered by spells (Brd Smn Blm Rdm Whm)

Number of TE's got seems to affect your ability to stagger
Number of TE's got seems to affect the color stagger you will get

Staggers /seem/ to just be a % chance to proc from any ws/spell/ja, with no pattern (eg sleep a mob and spam voke on it, first 10 may do nothing and the 11th proc).

Staggers should always be doable by people in your party (which fits with it being a % chance to proc).

What we dont know 100% yet is how TE's are affecting it, what the difference is between the color staggers in terms of drops, is it definatly just % chance on any ja/ws/spell and what does each specific KI from the TE's do.

Also the higher level mobs definatly can be staggered, ive dont it myself.