After an amphibious assault can overland attacking aircraft stay in newly captured territory?
Attacking aircraft can never land in a newly captured territory.  Being an amphibious assault does not change this rule. 

The fighters need to return to a territory that was friendly since the beginning of the turn.  Note that if the aircraft return to a land territory, it does not need to be the same one that they departed from.  Fighters can also choose to land on an aircraft carrier.  A single carrier can support two fighter planes.