In this answer, I am assuming that you are talking about the price that can be gotten by selling the chicken, not the price of producing the chicken.
If the price for which chicken can be sold goes down, the supply curve for eggs will shift to the left. In other words, the supply of eggs will go down, assuming that chickens and eggs are complements in production.
The reason for this is simple. If the price of chickens goes down, farmers will raise fewer chickens, all other things being equal. If farmers raise fewer chickens, there will also be fewer eggs laid. When that happens, the supply decreases and the supply curve moves to the left.