You cannot. But since NPCs cannot naturally hold such items, its likely someone (maybe you) added a custom item or something to its fitout.
Try telling the NPC to drop all its fitout items, and it should drop on the ground. Then just move it to your primary/secondary slot or into a backpack or something.
Until you claim the egg, it is 'owned' by NPCs, just do as Ellias said and place it in one of your hands and then refresh the equipment/inventory page and then click the newly-revealed [Claim] button