Ok, despite multiple wrong descriptions of how to colour your yoshi, which include naming your file after a colour, the colour of the file itself, or complete randomness, there actually is a real and definite way to colour your yoshi. There are seven different colours of yoshis: green, red, blue, orange, pink, black(ish) and white. The colour is determined by the time you leave it in the egg.

0-6 minutes: Green
6-9: Red
9-11: Blue
11-15: Orange
15-18: Pink
18-19: Black(ish)
19-20: White (the rarest)