Shifts to lower transition temperatures are observed for the freezing and melting of submicron-sized hexadecane droplets encapsulated within thin polystyrene shells. Supercooling of similar to 14 K is observed in the first-time cooling scans for the oil. We attribute this lowering predominantly to nucleation of the phase change originating from the oil/polymer/water interface. We obtain a rough estimate of the interfacial tension between the hexadecane oil droplet and the polystyrene of 14 mN/m assuming the Gibbs-Thomson relationship. Melting points for the hexadecane are 1-2 K below the bulk transition temperatures. This effect is connected with the surface/volume ratio of the capsules. Both the supercooling and the melting point depression approach the bulk-phase transition temperature when the sample is taken through multiple cool/heat cycles. The heating and cooling rates affects the number of cycles required before bulk-like behavior is observed. The thickness of the capsule wall is also observed to be critical to how many cooling cycles are required. Two hypotheses to explain this behavior are presented.