Aggregates of organic dyes that exhibit excitonic coupling have a wide array of applications, including medical imaging, organic photovoltaics, and quantum information devices. The optical properties of a dye monomer, as a basis of dye aggregate, can be modified to strengthen excitonic coupling. Squaraine (SQ) dyes are attractive for those applications due to their strong absorbance peak in the visible range. While the effects of substituent types on the optical properties of SQ dyes have been previously examined, the effects of various substituent locations have not yet been investigated. In this study, density functional theory (DFT) and time-dependent density functional theory (TD-DFT) were used to investigate the relationships between SQ substituent location and several key properties of the performance of dye aggregate systems, namely, difference static dipole (Δd), transition dipole moment (μ), hydrophobicity, and the angle (θ) between Δd and μ. We found that attaching substituents along the long axis of the dye could increase μ while placement off the long axis was shown to increase Δd and reduce θ. The reduction in θ is largely due to a change in the direction of Δd as the direction of μ is not significantly affected by substituent position. Hydrophobicity decreases when electron-donating substituents are located close to the nitrogen of the indolenine ring. These results provide insight into the structure–property relationships of SQ dyes and guide the design of dye monomers for aggregate systems with desired properties and performance.