Do you need 3-wire romex to feed bedroom ceiling box by electrical code?

No, you can feed it with a 2 wire Romex + ground. It depends on what is mounted on the ceiling. If it is just a light all you need is 2 wire + ground Romex. However if it is a fan/light and you want to control each one independant of the other you will need to use 3 wire Romex + ground. This is of course if you have 2 seperate switches. You would then connect the red wire to the blue light wire and the black wire to the black fan wire. If you use 2 wire Romex just connect the blue and black fan/light wire to the black wire in the ceiling box.