In biomedical and public health association studies, binary outcome variables may be subject to misclassification, resulting in substantial bias in effect estimates. The feasibility of addressing binary outcome misclassification in regression models is often hindered by model identifiability issues. In this paper, we characterize the identifiability problems in this class of models as a specific case of "label switching" and leverage a pattern in the resulting parameter estimates to solve the permutation invariance of the complete data log-likelihood. Our proposed algorithm in binary outcome misclassification models does not require gold standard labels and relies only on the assumption that outcomes are correctly classified at least 50% of the time. A label switching correction is applied within estimation methods to recover unbiased effect estimates and to estimate misclassification rates. Open source software is provided to implement the proposed methods. We give a detailed simulation study for our proposed methodology and apply these methods to data from the 2020 Medical Expenditure Panel Survey (MEPS).