Molecular excitons, which are excitations delocalized over multiple dyes in a wavelike manner, are of interest for a wide range of applications, including quantum information science. Numerous studies have templated a variety of synthetic dyes via a DNA scaffold to induce dye aggregation to create molecular excitons upon photoexcitation. Dye aggregate optical properties are critically dependent on relative dye geometry and local environment; therefore, an understanding of dye-dye and DNA-dye interactions is critical for advancing toward more complex DNA-dye systems. The extensively studied DNA Holliday junction (HJ) and less-studied double-crossover (DX) tile motif are fundamental test beds for designing complex and ultimately modular DNA-dye architectures. Here, we report the first study of single-linked squaraine dye aggregation and exciton delocalization on a larger and more stable (compared with the HJ) DX tile motif. We first highlight a few DNA-dye constructs that support single dyes and aggregates with distinct optical properties that are both tunable-through sample design, buffer conditions, and heat treatment-and robust to environment changes, including transfer to solid phase. Next, we assess several experimental and design considerations that demonstrate directed dye-driven assembly of a novel double-tile DNA configuration. Our results demonstrate that single-linked squaraine dyes templated to DX tiles provide a viable research path to design and evaluate dye aggregate networks that support exciton delocalization. We include herein the first report of exciton delocalization in the solid phase in a DNA-dye construct. Additionally, our findings indicate that dye aggregation impacts the assembly of the DNA-dye construct, and, in some cases, thereby cooperates with the DNA to determine a final robust system configuration. Finally, we show that a controlled annealing schedule can be employed to promote the homogeneous assembly of DNA-dye constructs. The findings in this study contribute to the understanding of DNA-dye systems and the relevant factors involved in their directed assembly to achieve specific constructs with desirable properties.