United Airlines does not stipulate limitations on the carriage of brachycephalic dogs such as Frenchies. His carriage was fully within the airline's terms.
I doubt whether the owners knew about the health risk to their puppy posed by flying.
United had banned carriage during summer for brachycephalic animals, and reinstated carriage in September I think it was. Heaven forbid a passenger being allowed to exercise a freedom and not knowing the risks being accepted, grab me a lawyer, it's somebodies fault, I can't possibly accept responsibility myself. That the owners possibly didn't know about the breeds medical issues is an indictment on a lot of pet owners who just purchase a fashion item or a toy for the kids because its cute. Breathing problems are endemic in the breed and not a result of backyard breeders. More often than not many require operations to enable them to breath. Read the following for some of the problems the breed has, they can't even reproduce without intervention - artificial insemination and cesarean birth being the norm.