Boost X

You may pay an additional X when you play this to get the boost effect.

If something has you "put a unit into play," that's different from "playing it" so you can't pay for or use a boost effect in that case. Likewise, if a unit with a boost enters play through any means other than playing it (such as being returned from Second Chances or Geiger or Pasternaak's max level abilities) then you can't use its boost. — Sirlin
Using Graveyard, Jurisdiction, and Vir Garbarean you can "play" a card from a zone other than your hand. You can still use boost when playing a card this way. — sharpo
If an opponent has Jail and you play a unit with a boost, you CAN pay for and use the boost. You do that immediately as you play the unit from your hand, then the boost effect happens and your unit goes to Jail. When it leaves Jail and arrives in play, you do not have a chance to pay for or use the boost a second time. — Sirlin