The growing interest in integrating liquid crystals (LCs) into flexible and miniaturized technologies brings about the need to understand the interplay between spatially curved geometry, surface anchoring, and the order associated with these materials. Here, we integrate experimental methods and computational simulations to explore the competition between surface-induced orientation and the effects of deformable curved boundaries in uniaxially and biaxially stretched nematic and smectic microdroplets. We find that the director field of the nematic LCs upon uniaxial strain reorients and forms a larger twisted defect ring to adjust to the new deformed geometry of the stretched droplet. Upon biaxial extension, the director field initially twists in the now oblate geometry and subsequently transitions into a uniform vertical orientation at high strains. In smectic microdroplets, on the other hand, LC alignment transforms from a radial smectic layering to a quasi-flat layering in a compromise between interfacial and dilatation forces. Upon removing the mechanical strain, the smectic LC recovers its initial radial configuration; however, the oblate geometry traps the nematic LC in the metastable vertical state. These findings offer a basis for the rational design of LC-based flexible devices, including wearable sensors, flexible displays, and smart windows.