You have stated that chickens and eggs are complements in production. This implies that they are produced together. If the number of chickens produced goes up, so does the number of eggs and vice versa.
As far as supply is concerned, the quantity produced is directly proportional to the price, as price increases or decreases, so does the quantity produced and supplied.
If the price of chickens goes down, producers would sell less chicken, and this would require less chicken to be produced. As eggs and chicken are complements in production, if the production of chicken is reduced so is the production of eggs. Therefore the supply curve of eggs moves to the left.
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.