I suspect you will notice a taste improvement, but not because of any special chemistry between a chicken and its own eggs. Rather, in order to connect the chicken and the egg, you will have to go either straight to the farmer, or possibly to a very small store that knows the farmer well. Further, the farmer will have to keep so few chickens that they know which ones laid which eggs, and generally be able to tell them apart. Imposing these conditions will lead to a very small operation, possibly a hobby farm of heritage breeds raised for love and fun, in which the chickens range free, play with children etc. And darn right you're going to end up with a better tasting dinner after you do that.
If you're able to source meat and eggs like this, as an experiment, you could easily bread meat from chicken A with eggs from chicken A, then another piece of meat from chicken A using eggs from chicken B, and what I think you would find is they taste the same - way better than the supermarket.
I think whoever suggested this just liked the thrill of the self-referential part, which I actually find a little gross.