Well, the egg.....the animal we know as the chicken today evolved from animals which laid eggs previous to that so the egg must have come first....What ever the answer to this one it's nearly always the man and not the woman in answer to another similar question.
The first recognisable bird, Archiopterix (sp) came into being when a 'dinosaur' reptile modified its scales into feathers. (All Creationists may now damn me to Hell). All other birds, including chickens followed on. As reptiles use eggs as their method of reproduction, the birds carried on using this method. Thus, the egg came before the chicken.
or on another note- assuming that chickens are bred from the indian running fowl (a wild bird which isnt a chicken but is a bit chickeny).....this was hunted for its meat, not its eggs, which are apparently a bit mank. so the chicken, in a culinary sense, came first, as it was the meat people bred the hens for, not their eggs. they came as a bonus