Shape memory polymers (SMPs) are polymers with the shape memory effect. The biodegradable SMPs are candidate materials for making biomedical devices and scaffolds for tissue engineering. Superparamagnetic iron oxide nanoparticles (SPIO NPs) have recently been reported to promote the osteogenic induction of human mesenchymal stem cells (hMSCs). In this study, we synthesized water-based biodegradable shape memory polyurethane (PU) as the main component of the 3D printing ink for fabricating bone scaffolds. The 3D printing ink contained 500 ppm of SPIO NPs to promote osteogenic induction and shape fixity, and it also contained polyethylene oxide (PEO) or gelatin for the improvement of printability. Scaffolds were printed by the microextrusion-based low-temperature fuse deposition manufacturing (LFDM) platform. Both PU-PEO and PU-gelatin ink showed excellent printability. Shape memory properties were evaluated in 50 °C air and 37 °C water. PU-PEO scaffolds showed better shape fixity and recovery than PU-gelatin scaffolds, while the shape memory properties in water were better than those in air. hMSCs were seeded for evaluation of bone regeneration. The proliferation of the hMSCs in PU/gelatin and PU/gelatin/SPIO scaffolds was greater than that in PU/PEO and PU/PEO/SPIO scaffolds, confirming the better compatibility of gelatin vs PEO as the viscosity enhancer of the ink. The gradual release of SPIO NPs from the scaffolds promoted the osteogenesis of seeded hMSCs. With SPIO in the scaffolds, the osteogenesis increased 2.7 times for PU/PEO and 1.5 times for PU/gelatin scaffolds based on the collagen content. Meanwhile, SPIO release from PU/PEO/SPIO scaffolds was faster than that from PU/gelatin/SPIO scaffolds at 14 days, consistent with the better osteogenesis observed in PU/PEO/SPIO scaffolds. We concluded that 3D printed PU scaffolds with shape memory properties, biodegradability, and osteogenic effect may be employed to the minimally invasive surgical procedures as customized-bone substitutes for bone tissue engineering.