Can BD Aquaman use Lunge during a Transporter Move and Attack action?

Short answer: yes.

Lon ganswer : yes. And if he does, he can use HSS, move X squares, then immediatly before he makes his close-combat attack move 2 squares, attack, and then move again of his speed minus X.

Medium answer: yes, because he meets the prerequisite and HSS allows you to make a close combat attack.

Final wrinkle answer: But remember that you wouldn't be able to use Lunge to break away, hit, and then use the rest of your HSS movement.
Since you can't use this version of HSS when adjacent, it can't be used to get the close combat attack that would activate Lunge.

Ok Here is what I meant... or let me ask the next obvious question:
So Aquaman moves X squares and then anounces his Close combat attack and moves two squares. Can he use CCE against the target and then continue to move after?

Lunge doesn't let you use any power that the characte could normally use under the given circumstance.

You can't use CCE normally when using the Move and Attack ability and you still can't use CCE just because Lunger is being unsed with the M&A ability. The M&A ability is just a watered down version of HSS and HSS is a power action that only grants a free close or ranged combat attack... and CCE is a power action, not a close or ranged combat attack.

No. CCE is a power action. During Hypersonic, a character gets to make a close combat ATTACK as a free action.

Lunge works because it works during ANY action that includes a CC Attack.