The stress is removed after completing the maneuver (see page 7 in the rulebook), but the maneuver is still completed even if you get blocked and have to back up.

What happens if I have a stress token. I reveal a green maneuver, but in executing it I fly into another ship. Do I not remove the stress token since I didn't execute the maneuver?

You do execute the maneuver. Overlap or not moving at all do not change that fact. During check pilot stress you remove one stress token.

Ahh. Thank you all very much.

So 1 stress token will always be removed when a green maneuver is revealed, after the move is executed.

So 1 stress token will always be removed when a green maneuver is revealed, after the move is executed.

A minor point of clarification - it doesn't actually matter what difficulty is revealed. The stress check is based entirely on what you execute. So if you reveal a white but some other effect changes it to green before you execute it, you will still remove a stress.