The Job Shop Scheduling Problem (JSSP) is a well-known NP-hard problem in combinatorial optimization, where the objective is to optimize job assignments across machines while minimizing a specific criterion such as makespan. Traditional mathematical and heuristic approaches struggle with scalability and handling complex precedence constraints. Recent advances in artificial intelligence, particularly deep reinforcement learning (DRL) and supervised learning, have shown promise but face challenges such as training instability and reliance on labeled data. To overcome these limitations, we propose SchedulExpert, a novel neural architecture based on a Mixture of Experts (MoE) framework with self-supervised learning. SchedulExpert integrates a Graph Attention Network (GAT) encoder, an attention-based routing mechanism, and multiple expert modules to enhance flexibility and efficiency in scheduling decisions. Experimental evaluations on benchmark JSSP instances demonstrate that SchedulExpert achieves competitive performance against state-of-the-art metaheuristic and neural-based methods, offering a scalable and effective solution for real-world scheduling challenges.