In your inventory under Manage Piloted, you should have options in the action drop-down to Set Controller. Any ships assigned to you as pilot that are the same type as the one you're actually flying may be set to NPC Controlled, which means they'll follow you around where ever you go. for example: You're flying a Y-Wing. You can have up to 11 other Y-Wings set to follow you.
The shuttles in the ships are squad size 3, while everything else in that list if I recall correctly are squad size 12. Someone feel free to correct if I'm wrong.
If you are a race that cannot fly something due to restrictions, you can not stick an NPC in the cockpit to get around that. There is a different form of NPC Transport that is like taking a space bus for that, found in starports.
As for droids, they can't tell the ship to fly for you either, but having one in the cockpit while you're flying a ship lets you leave the room and they'll make sure the ship continues going where you told it.