In this paper, we examine the security aspects of two-way untrusted relaying networks (TWURNs) through the lens of physical layer security (PLS). These networks involve the exchange of confidential information between two multiple-antenna base stations (BSs) with the aid of unmanned aerial vehicle (UAV)-enabled untrusted relays equipped with a single antenna. Given the shortcomings observed in existing single untrusted relay selection methods, we introduce an enhanced approach grounded in the min-min criterion (MMC) based on received signal strength. Additionally, we present a novel closed-form expression for the secrecy outage probability (SOP), a fundamental metric within the domain of PLS, derived from the cumulative distribution function (CDF) and probability density function (PDF) of received signal strengths. Through simulations, we substantiate the validity of our analytical findings, demonstrating the superior efficacy of the proposed MMC compared to prevailing schemes.