For investigations into fate specification and cell rearrangements in live images of preimplantation embryos, automated and accurate 3D instance segmentation of nuclei is invaluable; however, the performance of segmentation methods is limited by the images' low signal-to-noise ratio and high voxel anisotropy and the nuclei's dense packing and variable shapes. Supervised machine learning approaches have the potential to radically improve segmentation accuracy but are hampered by a lack of fully annotated 3D data. In this work, we first establish a novel mouse line expressing near-infrared nuclear reporter H2B-miRFP720. H2B-miRFP720 is the longest wavelength nuclear reporter in mice and can be imaged simultaneously with other reporters with minimal overlap. We then generate a dataset, which we call BlastoSPIM, of 3D microscopy images of H2B-miRFP720-expressing embryos with ground truth for nuclear instance segmentation. Using BlastoSPIM, we benchmark the performance of five convolutional neural networks and identify Stardist-3D as the most accurate instance segmentation method across preimplantation development. Stardist-3D, trained on BlastoSPIM, performs robustly up to the end of preimplantation development (> 100 nuclei) and enables studies of fate patterning in the late blastocyst. We, then, demonstrate BlastoSPIM's usefulness as pre-train data for related problems. BlastoSPIM and its corresponding Stardist-3D models are available at: blastospim.flatironinstitute.org.