Bias and fairness of biometric algorithms have been key topics of research in recent years, mainly due to the societal, legal and ethical implications of potentially unfair decisions made by automated decision-making models. A considerable amount of work has been done on this topic across different biometric modalities, aiming at better understanding the main sources of algorithmic bias or devising mitigation measures. In this work, we contribute to these efforts and present the first study investigating bias and fairness of sclera segmentation models. Although sclera segmentation techniques represent a key component of sclerabased biometric systems with a considerable impact on the overall recognition performance, the presence of different types of biases in sclera segmentation methods is still underexplored. To address this limitation, we describe the results of a group evaluation effort (involving seven research groups), organized to explore the performance of recent sclera segmentation models within a common experimental framework and study performance differences (and bias), originating from various demographic as well as environmental factors. Using five diverse datasets, we analyze seven independently developed sclera segmentation models in different experimental configurations. The results of our experiments suggest that there are significant differences in the overall segmentation performance across the seven models and that among the considered factors, ethnicity appears to be the biggest cause of bias. Additionally, we observe that training with representative and balanced data does not necessarily lead to less biased results. Finally, we find that in general there appears to be a negative correlation between the amount of bias observed (due to eye color, ethnicity and acquisition device) and the overall segmentation performance, suggesting that advances in the field of semantic segmentation may also help with mitigating bias.