Hydrated salt phase change materials (PCMs) suffer from high supercooling and low thermal conductivity, limiting their solar thermal storage efficiency. To address these issues, we developed a novel shaped composite PCM via a synergistic MgO/EG co-modification strategy. A binary eutectic of magnesium sulfate heptahydrate (MSH) and magnesium nitrate hexahydrate (MNH) (molar ratio 6:4) served as the matrix. Magnesium oxide (MgO) acted as a nucleating agent, while expanded graphite (EG) formed a high-conductivity network. The composite, fabricated by melt mixing and cold pressing, exhibited dramatically reduced supercooling 1.71°C and a prolonged exothermic plateau 15 min. With only 15 wt% EG, thermal conductivity reached 6.86 W/(m·K) 21.24 times that of the pure salt with effective leakage prevention. The material maintained a high melting enthalpy of 104.4 J/g and excellent stability over 200 cycles. This work offers a promising, reliable composite for advanced solar thermal storage.