Human induced pluripotent stem cells (hiPSCs), human embryonic stem cells (hESCs) and human umbilical cord mesenchymal stem cells (hUCMSCs) are exciting cell sources for use in regenerative medicine. There have been no reports on long hydrogel fibers encapsulating stem cells inside an injectable calcium phosphate cement (CPC) scaffold for bone tissue engineering. The objectives of this study were: (1) to develop a novel injectable CPC construct containing hydrogel fibers encapsulating cells for bone engineering, and (2) to investigate and compare cell viability, proliferation and osteogenic differentiation of hiPSC-MSCs, hESC-MSCs and hUCMSCs in injectable CPC. The pastes encapsulating the stem cells were fully injectable under a small injection force, and the injection did not harm the cells, compared with non-injected cells (p > 0.1). The mechanical properties of the stem cell-CPC construct were much better than those of previous injectable polymers and hydrogels for cell delivery. The hiPSC-MSCs, hESC-MSCs and hUCMSCs in hydrogel fibers in CPC had excellent proliferation and osteogenic differentiation. All three cell types yielded high alkaline phosphatase, runt-related transcription factor, collagen I and osteocalcin expression (mean ± SD; n = 6). Cell-synthesized minerals increased substantially with time (p < 0.05), with no significant difference among the three types of cells (p > 0.1). Mineralization by hiPSC-MSCs, hESC-MSCs and hUCMSCs in CPC at 14 d was 13-fold that at 1 d. In conclusion, all three types of cells (hiPSC-MSCs, hESC-MSCs and hUCMSCs) in a CPC scaffold showed high potential for bone tissue engineering, and the novel injectable CPC construct with cell-encapsulating hydrogel fibers is promising for enhancing bone regeneration in dental, craniofacial and orthopedic applications.