Dye molecules, arranged in an aggregate, can display excitonic delocalization. The use of DNA scaffolding to control aggregate configurations and delocalization is of research interest. Here, we applied Molecular Dynamics (MD) to gain an insight on how dye–DNA interactions affect excitonic coupling between two squaraine (SQ) dyes covalently attached to a DNA Holliday junction (HJ). We studied two types of dimer configurations, i.e., adjacent and transverse, which differed in points of dye covalent attachments to DNA. Three structurally different SQ dyes with similar hydrophobicity were chosen to investigate the sensitivity of excitonic coupling to dye placement. Each dimer configuration was initialized in parallel and antiparallel arrangements in the DNA HJ. The MD results, validated by experimental measurements, suggested that the adjacent dimer promotes stronger excitonic coupling and less dye–DNA interaction than the transverse dimer. Additionally, we found that SQ dyes with specific functional groups (i.e., substituents) facilitate a closer degree of aggregate packing via hydrophobic effects, leading to a stronger excitonic coupling. This work advances a fundamental understanding of the impacts of dye–DNA interactions on aggregate orientation and excitonic coupling.