Single nanostructures embedded within nanowires (NWs) represent one of the most promising technologies for applications in quantum photonics. However, fabrication imperfections and etching-induced defects are inevitable for top-down fabrications, whereas self-assembly bottom-up approaches cannot avoid the difficulties of its stochastic nature and are limited to restricted heterogeneous material systems. Here we demonstrate the versatile self-assembly of single "square" quantum rings (QR) on the sidewalls of gold-free GaAs NWs for the first time. By tuning the deposition temperature, As overpressure and amount of gallium-droplets, we were able to control the density and morphology of the structure, yielding novel single quantum dots, QR, coupled QRs, and nano-antidots. A proposed model based on a strain-driven, transport-dependent nucleation of gallium droplets at high temperature accounts for the formation mechanism of these structures. We achieved a single-QR-in-NW structure, of which the optical properties were analyzed using micro-photoluminescence at 10 K and a spatially resolved cathodoluminescence technique at 77 K. The spectra show sharp discrete peaks; of these peaks, the narrowest linewidth (separation) was 578 μeV (1-3 meV), reflecting the quantized nature of the ring-type electronic states.