It would be the beginning of the Movement Phase for both, yes.
If fighter A used Combat Intuition, in the beginning of the Movement Phase (before other units) it would move (set a new flight path) and select a target, Fighter B. Fighter B would then choose whether to return fire or not.
If Fighter B does return fire, make the engagement roll, then make the attacks and apply the damage for both units immediately.
If Fighter B does not return fire, make the engagement roll, then Fighter A attacks and applies damage. If Fighter B is destroyed, it cannot make an attack in the Combat Phase that turn. If Fighter B is not destroyed, it can attack in the Combat Phase (possibly affected by any critical hits or other effects from Fighter A's attack).