Homelessness among US veterans remains a critical public health challenge, yet risk prediction offers a pathway for proactive intervention. In this retrospective prognostic study, we analyzed electronic health record (EHR) data from 4,276,403 Veterans Affairs patients during a 2016 observation period to predict first-episode homelessness occurring 3-12 months later in 2017 (prevalence: 0.32-1.19%). We constructed static and time-varying EHR representations, utilizing clinician-informed logic to model the persistence of clinical conditions and social risks over time. We then compared the performance of classical machine learning, transformer-based masked language models, and fine-tuned large language models (LLMs). We demonstrate that incorporating social and behavioral factors into longitudinal models improved precision-recall area under the curve (PR-AUC) by 15-30%. In the top 1% risk tier, models yielded positive predictive values ranging from 3.93-4.72% at 3 months, 7.39-8.30% at 6 months, 9.84-11.41% at 9 months, and 11.65-13.80% at 12 months across model architectures. Large language models underperformed encoder-based models on discrimination but showed smaller performance disparities across racial groups. These results demonstrate that longitudinal, socially informed EHR modeling concentrates homelessness risk into actionable strata, enabling targeted and data-informed prevention strategies for at-risk veterans.