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By explaining how humans would solve a given task, human rationales can provide strong learning signal for neural language models (LMs). Explanation regularization (ER) aims to improve LM generalization by pushing the LM's machine rationales (Which input tokens did the LM focus on?) to align with human rationales (Which input tokens would humans focus on?). Though prior works primarily study ER via in-distribution (ID) evaluation, out-of-distribution (OOD) generalization is often more critical in real-world scenarios, yet ER's effect on OOD generalization has been underexplored. In this paper, we introduce ER-Test, a framework for evaluating ER models' OOD generalization along three dimensions: unseen dataset tests, contrast set tests, and functional tests. Using ER-Test, we extensively analyze how ER models' OOD generalization varies with different ER design choices. Across two tasks and six datasets, ER-Test shows that ER has little impact on ID performance but can yield large OOD performance gains. Also, we find that ER can improve OOD performance even with limited rationale supervision. ER-Test's results help demonstrate ER's utility and establish best practices for using ER effectively.