The free showers are airside in the main terminal building on the 4th floor. If you want to exit the airport and visit the capsule hotel, go to Seoul, CGV movie theater or even the Korean spa in the basement of the airport.... you cannot use those free showers. You'd have to stay inside the airport as a transit passenger.
Try the Korean spa in the basement for a shower if you don't want to go into Seoul. If you do go into Seoul, I highly recommend Siloam Sauna for a shower and sauna experience. Even the food there is gooooood!