We introduce a novel approach called the Bayesian Jackknife empirical likelihood method for analyzing survey data obtained from various unequal probability sampling designs. This method is particularly applicable to parameters described by U-statistics. Theoretical proofs establish that under a non-informative prior, the Bayesian Jackknife pseudo-empirical likelihood ratio statistic converges asymptotically to a normal distribution. This statistic can be effectively employed to construct confidence intervals for complex survey samples. In this paper, we investigate various scenarios, including the presence or absence of auxiliary information and the use of design weights or calibration weights. We conduct numerical studies to assess the performance of the Bayesian Jackknife pseudo-empirical likelihood ratio confidence intervals, focusing on coverage probability and tail error rates. Our findings demonstrate that the proposed methods outperform those based solely on the jackknife pseudo-empirical likelihood, addressing its limitations.

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