High-throughput 3D quantitative phase imaging (QPI) in flow cytometry enables label-free, volumetric characterization of individual cells by reconstructing their refractive index (RI) distributions from multiple viewing angles during flow through microfluidic channels. However, current imaging methods assume that cells undergo uniform, single-axis rotation, which require their poses to be known at each frame. This assumption restricts applicability to near-spherical cells and prevents accurate imaging of irregularly shaped cells with complex rotations. As a result, only a subset of the cellular population can be analyzed, limiting the ability of flow-based assays to perform robust statistical analysis. We introduce OmniFHT, a pose-free 3D RI reconstruction framework that leverages the Fourier diffraction theorem and implicit neural representations (INRs) for high-throughput flow cytometry tomographic imaging. By jointly optimizing each cell's unknown rotational trajectory and volumetric structure under weak scattering assumptions, OmniFHT supports arbitrary cell geometries and multi-axis rotations. Its continuous representation also allows accurate reconstruction from sparsely sampled projections and restricted angular coverage, producing high-fidelity results with as few as 10 views or only 120 degrees of angular range. OmniFHT enables, for the first time, in situ, high-throughput tomographic imaging of entire flowing cell populations, providing a scalable and unbiased solution for label-free morphometric analysis in flow cytometry platforms.