In my fantasy world, unicorns have pearly horns, teeth, and bones. I didn't write it, but I am hoping to write a fanfiction in which it is explained.
Nacre is composed of hexagonal platelets of aragonite (a form of calcium carbonate) 10–20 µm wide and 0.5 µm thick arranged in a continuous parallel lamina. Depending on the species, the shape of the tablets differ; in Pinna, the tablets are rectangular, with symmetric sectors more or less soluble. Whatever the shape of the tablets, the smallest units they contain are irregular rounded granules. These layers are separated by sheets of organic matrix (interfaces) composed of elastic biopolymers (such as chitin, lustrin and silk-like proteins). This mixture of brittle platelets and the thin layers of elastic biopolymers makes the material strong and resilient, with a Young's modulus of 70 GPa (when dry)...
Nacre appears iridescent because the thickness of the aragonite platelets is close to the wavelength of visible light. These structures interfere constructively and destructively with different wavelengths of light at different viewing angles, creating structural colours.
Your unicorns lay down microscopic plates of calcium carbonate and biopolymer, ocating these various structures. Horns, teeth and bones are all made of different stuff but just as the molluscs coat their shellstuff with nacre, so too your unicorns coat all their various hard parts with their nacre equivalent.
Iridescence is caused by a diverse range of microstructures, so there is no reason they cannot. Your best bet is to have photonic crystals as they are fairly easy to make biologically (they pop up all over the animal kingdom). They are structural more than material so you can make them with a wide range of materials. Teeth and horns are fine and in theory bone could be used to make photonic crystals so you could have it in the bones as well.
The real question you need to answer is why they have these structures. Horns could be for simple sexual display, everywhere else becomes problematic however.