One of the most important areas of research in the biology and genetics of farmed animals is one of identification of genes controlling the expression of traits with practical importance for animal breeding. For most of these characteristic features, the wide variation in gene expression in specific loci, which are called quantitative trait loci (QTL), is typical. Eggs have been researched for decades due to their importance for the reproduction of birds, as well as for its widespread use in pharmaceutical, cosmetic, and food industries. Breeding hens and crosslines are a necessary step for producing eggs with the desired quality. The results of this work are recommended for use to create systems of molecular markers for the marker selection of layers and obtain new lines and cross hens with larger mass eggs. Compared to the existing conventional systems of selecting layers on this basis, this will eliminate the assessment of the genotype of male progeny, which will significantly reduce breeding time. The system of markers will appear as a set of primers for detecting gene alleles that have a significant impact on the characteristics as above. The use of the molecular markers of high-performance systems for direct selection based on domestic chicken eggs should lead to substantial progress in biotechnology poultry and help avoid having to purchase similar systems from outside the country. The association of the condensin NCAPG gene with the egg traits of domestic chicken has been studied. Associations of the SNP alleles of the rs14491030 marker localized in exon 8 of the NCAPG gene with the trait “the weight eggs,” p < 0.001, as well as with the elastic deformation of the egg shell, p < 0.026, have been found. It has been found that a single nucleotide nonsynonymous A → G substitution leads to a significant increase in egg weight. The marker SNP rs14491030 with the observed significant effect on the egg weight trait can be recommended for use in breeding laying hens. The calculations of the relative fitness of genotypes of the SNP rs14491030 marker suggest a natural selection for heterozygotes. The results obtained are discussed in connection with the role of the canonical condensin complex in the compaction of chromatin and segregation of chromosomes.