Foundation models, pre-trained on large image datasets and capable of capturing rich feature representations, have recently shown potential for zero-shot image registration. However, their performance has mostly been tested in the context of rigid or less complex structures, such as the brain or abdominal organs, and it remains unclear whether these models can handle more challenging, deformable anatomy. Breast MRI registration is particularly difficult due to significant anatomical variation between patients, deformation caused by patient positioning, and the presence of thin and complex internal structure of fibroglandular tissue, where accurate alignment is crucial. Whether foundation model-based registration algorithms can address this level of complexity remains an open question. In this study, we provide a comprehensive evaluation of foundation model-based registration algorithms for breast MRI. We assess five pre-trained encoders, including DINO-v2, SAM, MedSAM, SSLSAM, and MedCLIP, across four key breast registration tasks that capture variations in different years and dates, sequences, modalities, and patient disease status (lesion versus no lesion). Our results show that foundation model-based algorithms such as SAM outperform traditional registration baselines for overall breast alignment, especially under large domain shifts, but struggle with capturing fine details of fibroglandular tissue. Interestingly, additional pre-training or fine-tuning on medical or breast-specific images in MedSAM and SSLSAM, does not improve registration performance and may even decrease it in some cases. Further work is needed to understand how domain-specific training influences registration and to explore targeted strategies that improve both global alignment and fine structure accuracy. We also publicly release our code at \href{https://github.com/mazurowski-lab/Foundation-based-reg}{Github}.