In this paper, we investigate a problem of minimizing total energy consumption for secure federated learning (FL) over wireless edge networks. To address the high computational cost and privacy challenges in conventional FL with neural networks (NN) for resource-constrained users, we propose a novel FL with hyperdimensional computing and differential privacy (FL-HDC-DP) framework. In the considered model, each edge user employs hyperdimensional computing (HDC) for local training, which replaces complex neural updates with simple hypervector operations, and applies differential privacy (DP) noise to protect transmitted model information. We optimize the total energy of computation and communication under both latency and privacy constraints. We formulate the problem as an optimization that minimizes the total energy of all users by jointly allocating HDC dimension, transmission time, system bandwidth, transmit power, and CPU frequency. To solve this problem, a sigmoid-variant function is proposed to characterize the relationship between the HDC dimension and the convergence rounds required to reach a target accuracy. Based on this model, we develop two alternating optimization algorithms, where closed-form expressions for time, frequency, bandwidth, and power allocations are derived at each iteration. Since the iterative algorithm requires a feasible initialization, we construct a feasibility problem and obtain feasible initial resource parameters by solving a per round transmission time minimization problem. Simulation results demonstrate that the proposed FL-HDC-DP framework achieves up to 83.3% total energy reduction compared with the baseline, while attaining about 90% accuracy in approximately 3.5X fewer communication rounds than the NN baseline.