Domain Adaptation (DA) is crucial for robust deployment of medical image segmentation models when applied to new clinical centers with significant domain shifts. Source-Free Domain Adaptation (SFDA) is appealing as it can deal with privacy concerns and access constraints on source-domain data during adaptation to target-domain data. However, SFDA faces challenges such as insufficient supervision in the target domain with unlabeled images. In this work, we propose a Segment Anything Model (SAM)-guided Reliable Pseudo-Labels method for SFDA (SRPL-SFDA) with three key components: 1) Test-Time Tri-branch Intensity Enhancement (T3IE) that not only improves quality of raw pseudo-labels in the target domain, but also leads to SAM-compatible inputs with three channels to better leverage SAM's zero-shot inference ability for refining the pseudo-labels; 2) A reliable pseudo-label selection module that rejects low-quality pseudo-labels based on Consistency of Multiple SAM Outputs (CMSO) under input perturbations with T3IE; and 3) A reliability-aware training procedure in the unlabeled target domain where reliable pseudo-labels are used for supervision and unreliable parts are regularized by entropy minimization. Experiments conducted on two multi-domain medical image segmentation datasets for fetal brain and the prostate respectively demonstrate that: 1) SRPL-SFDA effectively enhances pseudo-label quality in the unlabeled target domain, and improves SFDA performance by leveraging the reliability-aware training; 2) SRPL-SFDA outperformed state-of-the-art SFDA methods, and its performance is close to that of supervised training in the target domain. The code of this work is available online: https://github.com/HiLab-git/SRPL-SFDA.