Blue French Bulldogs are against the AKC standard and are disqualifiable in the AKC conformation ring. However they can be properly registered with grey by the AKC on their registration certificates. Furthermore there are alternative registries that accept Colored French Bulldogs.
Strict anti-color enthusiasts like to say that French Bulldogs of DQ (Disqualifiying) Color, like blues, are less healthy or more prone to health issues. This is absolutely FALSE! Traditional black/brindle/seal have the same possibility of having the same health issues as nontraditional colors like blue/brown/cream and so forth.
Blue is a recessive of the French Bulldog breeds Color genetics. Any breed that has the color "dominant black" within it's genetics can have blue born within it. Why? Because it can be created by a mutation of Chromosome 25 which dilutes pigmentation. In the case of the blues they are a dilute of black. This is why you can see blue Rottweilers, Dobermans, Miniature Pinschers and so much more. Purebred, but a genetic anomaly. This mutation has never been proven to cause any health problems in itself. Why? Because canine coat color genetics rarely affect any actual health matters with dogs. I say rarely because there are things like with Merle that can be deadly.
Dilutes and Colour Dilution Alopecia :
There is a common misconception that dilutes are in some way naturally sickly - this is not in fact the case. The dilution gene does impair the ability of the cells to make pigment, but only in that it causes the pigment that is made to be less intense. As with most recessives, the dilute allele is in some way faulty", but it is only faulty in its ability to produce full-strength eumelanin.The ability or inability of the cells to produce full-strength eumelanin does not affect the health of the dog, simply its color.
That said, the idea of dilutes as unhealthy most likely has its foundations in Colour Dilution Alopecia. This is an apparently genetic disease causing hair loss and skin problems. A dog with this disorder will typically appear "mangy" and have partial hair loss. It is usually reported from blue dogs, but presumably it affects isabella dogs too (diluted livers). Any colour can carry CDA or be homozygous for it, but only blues and isabellas will have symptoms.
CDA does not occur on all dilutes and its frequency varies between breeds. It is particularly common in Dobermans, occurring in up to 80% of dilute Doberman dogs. Dilutes in other species such as mice are caused by the same gene, and yet CDA is not known in these, implying it is not an unavoidable consequence of dilution. It is thought that CDA may be caused by a specific dilution gene - labelled dl. Just as there are various different b alleles that all cause the liver color (pheno typically the same, so only distinguishable through genetic testing), it is probable that there are a number of different d alleles as well, and only one of these causes CDA.What this means is that CDA is most likely caused by a recessive allele but could theoretically be bred out of most lines by careful breeding and genetic testing.
The same problem can also occur (albeit rarely) on black or liver dogs, and is known as Black Hair Follicular Dysplasia. It affects black/liver hairs only, leaving all other hairs as normal. Because this condition is so rare, it often goes undiagnosed. Unfortunately for dogs with genetic hair loss conditions, there is no cure, although these conditions do not generally cause the dog to be itchy or uncomfortable and so are mostly harmless.
Both the liver gene and the blue gene are recessive, and thus can be carried by practically any dog. The "mouse" color that is mentioned in the standard appears to refer to both blue dogs and dogs that are expressing both blue and liver at the same time, which is in other breeds known by such names as grey, fawn, Isabella, lavender, and lilac. However, the lilac coloration is seen least often of the three due to the need of the dog being homozygous recessive for both the liver and blue genes. As such, it's quite possible that "mouse" refers only to blue and the liver+blue lilac is completely ignored by the standard. This is what is implied by the French Bulldog Club of America. This amuses me somewhat since lilac can appear quite similar to fawn in color and since the standard allows for light pigment in light colored dogs, it's possible that a lilac could be considered acceptable.
Once you have a pair of blues or a pair that carry the gene, when bred together they can produce blue pups. It's fairly easy to "create" blue once it's within a pedigree since it can be DNA tested for. Most breeders breed for color alone with no regard for health testing or care for structure type so the colored French Bulldogs don't even look like French Bulldogs sometimes which is the problem most anti-color enthusiasts have with colored breeders. Thankfully there are French Bulldog breeders who are striving for to produce better and better quality "alternative" colotred French Bulldogs as time goes on.
Deafness is a huge concern with the French Bulldog breed because of the use of English White Terriers (a breed that went extinct because of deafness) to create the French Bulldog breed at it's origins.
Also there is no known link to deafness because of coat color. There is no scientific proof as to what causes deafness. There is no "deafness gene" that has been mapped to check before breeding. In fact, there are on theories, again no solid proof. The best theory is that lack of pigmentation in the inner ear canal is a link and this is why you see a higher incidence of white headed, white eared dogs (again their pattern or markings not color) with hearing issues. Yes you can still have a traditional "show" marked dog that is deaf with dark inner ear canal. That is why deafness is such a mystery.
Of all the colors and markings, the most seen to be deaf in both ears have been traditional black/brindle with traditional "show" markings and brown eyes. Goes against what a lot of people believe huh? There have been some colored, mismarked, and/or blue eyed (one or both) that have resulted unilaterally or bilaterally deaf so allcolors are not immune from deafness. The numbers of research on deafness speak volumes to disprove the opinions of colors having more issue with deafness this is factual based information, not opinion.