Boone - if you've already done all of the quests/situations that gets you "trust points" with him, you're locked out. If you're pretty close to the end of the game/done tons of stuff already, this is most likely going to be the case (locked out).
Arcade - pretty sure as long as you haven't reached a certain point in the main plot(close to the end), you can do his anytime. His is just triggered by taking him to certain places and he'll say stuff. There are also some quest-specific situations for gaining trust points that you
can be missed/locked out of, but there's enough others to do without those. He won't trigger his quest until a certain point in the game even after you get the favor points. I think if you do YesMan you don't even need the trust points, but never went that way myself so no personal experience.
Raul - there was a bug with Ranger Andy (and maybe the others) where you couldn't trigger Raul's quest if you'd already talked to Randy/the others without Raul present (Raul's dialogue would then not trigger). This was fixed in one of the patches, but with this game, whether the fix actually works for you is always up in the air.

Cass - If you've killed the Van Graffs or McLafferty, her quest may stall/not be completable since confronting them are vital to her quest. Other than that, should be able to do it any time.
Veronica - if you've destroyed the BoS bunker already (no npc's in there) I'd make the guess you'd be locked out.

Other than that I'm not sure. She's always been a bit wacky for me...she supposedly has quite a few triggers and sometimes will trigger even without those under certain circumstances, so hard to say for sure.
Lily - no idea, don't care.
EDE - with the patches I think he's doable no matter what (should be, if not glitched), except if the BoS bunker is gone.
Edit: In short, there's no single/simple answer. Each companion is a bit different in regards to whether you can do their quest "at anytime" - even if you can do them "anytime", there may only be one specific path left to you if you wait a long time.
Personally I'd advise you to go through the quest guides on the wiki for each companion quest ... it'll list known triggers/combos, possible bugs/glitches, etc. http://fallout.wikia.com/wiki/Fallout:_New_Vegas_companions