Why do skin and nail go blue in cyanosis?
The skin appears blue due to the diffraction of light through the skin. Blood is either bright red (sufficiently oxygenated) or dark red. "The blue appearance of surface veins is caused mostly by the scattering of blue light away from the outside of venous tissue if the vein is at 0.5 mm deep or more. Veins and arteries appear similar when skin is removed and are seen directly" 1. The color of blood is due to the difference in color between deoxyhemoglobin and oxyhemoglobin. 

Let's consider the "blue fugates" of Kentucky. The ancestors of this family have different shades of blue skin tones. Dr. Madison Cawein discovered that the family had a high concentration of FE$^{3+}$ brown methemoglobin. This form is useless to the body so the enzyme diaphorase reduces the methemoglobin to FE$^{2+}$ (hemoglobin that can carry oxygen) [2].

Basically the skin color comes down to light diffraction not the blood changing from red to blue.


Cyanosis is the appearance of a blue or purple coloration of the skin or mucous membranes due to the tissues near the skin surface having low oxygen saturation [3]. Cyanosis is a clinical sign diagnosed visually. Cyanosis is an indication for blood gas testing. 

References:


http://en.wikipedia.org/wiki/Venous_blood
http://courses.chem.indiana.edu/c122/documents/Experiment10_BloodandHemoglobin.pdf
http://en.wikipedia.org/wiki/Cyanosis
http://www.webmd.com/lung/arterial-blood-gases