|"The outer piece of my CD player fell off".|
I doubt the piece just fell off. If it was knocked off then it may be broken.
That said, many drives are sold with two colors of faces. That means they are easy to change by the user. You just need to look at the parts.
From your description it seems the errant part is the drive facing. Either way, both are generally removable. You may need to remove the drive from the case if it is the drive front. The tray fronts usually just slide on from the top. You need to support the tray or you will break it.
One other hint. Try installing the drive facing while the computer is off and you have manually ejected the tray. Look at the metal parts for notches that plastic protrusions snap into.