Generally you'd want an ADC who either
1. Has range
2. Has early game damage
3. Has CC
Basically Support Anivia works best with Ashe, Vayne, or Varus
Anivia + Ashe = Enemy Team has no movespeed (Frostshot OP)
Anivia + Vayne = Tons of True Damage as vayne will probably hit at least 3 aa
Anivia + Varus = Well. Um. Op
Not so good with twitch as his passive got nerfed QQ
Squishy or not doesn't matter as you can peel so hard with your Q, W and R AND your actives
However, Anivia is so underplayed nobody knows her rebirth passive, so make sure your ADC knows about your passive and make use of it. (A.K.A. killing their ADC who should be focusing on the egg and not the adc)

Early Game, you should try to hit as MANY stuns as you can, and only wait for the best situations as you can literally only shoot 3 of them at level 1 and you are so mana hungry. If you manage to hit all of them, their adc should be really low, and you should have the advantage

Mid Game, you basically peel for the heck of it. Only use your ult when
1. You get a gank
2. ADC can get a clear kill if you do damage
3. Enemy Team is going for your ADC

Late Game, you should pretty much be the most annoying thing ever: SLOWING, STUNNING, BLOCKING
and the best part, if you get a WOTA, you'll heal if they chase you.
Even if they chase you, you just stun, ult, then wall, then KEEP RUNNING as you do have boots of mobility since anivia lacks mobility