Hybrid controlled trials (HCTs), which augment randomized controlled trials (RCTs) with external controls (ECs), are increasingly receiving attention as a way to address limited power, slow accrual, and ethical concerns in clinical research. However, borrowing from ECs raises critical statistical challenges in estimation and inference, especially for binary outcomes where hidden bias is harder to detect and estimands such as risk difference, risk ratio, and odds ratio are of primary interest. We propose a novel framework that combines doubly robust estimators for various estimands under covariate shift of ECs with conformal selective borrowing (CSB) to address outcome incomparability. CSB uses conformal inference with nearest-neighbor-based conformal scores and their label-conditional extensions to perform finite-sample exact individual-level EC selection, addressing the limited information in binary outcomes. To ensure strict type I error rate control for testing treatment effects while gaining power, we use a Fisher randomization test with the CSB estimator as the test statistic. Extensive simulations demonstrate the robust performance of our methods. We apply our method to data from CALGB 9633 and the National Cancer Database to evaluate chemotherapy effects in Stage IB non-small-cell lung cancer patients and show that the proposed method effectively mitigates hidden bias introduced by full-borrowing approaches, strictly controls the type I error rate, and improves the power over RCT-only analysis.