Smitty0305 wrote: "When using their turbo-boosters they may move up to 24' in the movement phase"
You pretty much answered your own question here. The turbo-boost is made in the Movement phase.
Smitty0305 wrote:"A unit using turbo boosting must end its move 18' away from its starting point to claim this cover save"
Fixed the rules quote for you. It's not "end its turn" it's "end its move", so to get the cover save you must move more than 18" in the Movement phase. It has to take place in that phase because you can't shoot, so you can't add any distance from running, and you can't assault, so you can't add any distance from moving in the assault phase.