Description:
There is a single study done that proposed the red egg mutation in Coturnix quail, found at a New Hampshire University. It has not been referenced in a lab setting further, and no pictures exist from the study. It is not believed to be in public hands right now, but has not been studied further.
The general color classification for quail eggs is by their ground/base color, their spot color and size, and inner shell color. The red eggshell gene causes the typical tan base color to turn a light salmon color, and changes the spots from a dark brown to a reddish bay tone. It does not affect the spot size. The inner shell coloration has not been described, and likely stays the same as typical wildtype eggshell color. It is unknown how visible this red egg gene is, it is possible that it is in backyard flocks and just gone unnoticed.
It was not linked to any plumage color, and no health or shell abnormalities were described in the 35 generations they were tested. It is likely that the gene could appear with any plumage color. There also is no way to realistically see the gene in males except for test breeding and individually studying the female offspring.
The Red Egg gene was test bred with a "light blue eggshell" recessive gene in their flock (likely Celadon), and found to be recessive to the Celadon. The red egg gene could not be spotted whatsoever if the bird was homozygous for the blue egg gene. No further studies were done to see if this was because they were linked or if on different loci.
Breeding with Red Eggshell
Note: Red Egg here refers to the bird itself that lays the red egg. It does not refer to the egg. For example, the 50% Red Splits means half of the resulting birds will carry one copy of the gene, not that it will lay half red eggs and half red splits.
Red Egg x Red Egg = 100% Red Egg
Red Egg x Wildtype Egg = 100% Red Splits
Red Egg x Red Splits = 50% Red Eggs, 50% Red Splits
Red Splits x Red Splits = 25% Red Eggs, 50% Red Splits, 25% Wildtype Eggs
Red Splits x Wildtype Egg = 50% Red Splits, 50% Wildtype Egg
References & Further Reading
J. W. HARDIMAN, W. M. COLLINS, W. E. URBAN, Red Egg-Shell Color: A Dominant Mutation in Japanese Quail, Journal of Heredity, Volume 66, Issue 3, May 1975, Pages 141–143, https://doi.org/10.1093/oxfordjournals.jhered.a108596