Quantifying the detection likelihood of faint peaks in interferometric data through jackknifing: Test application on finding $z>10$ galaxy candidates

False-positive emission-line detections bias our understanding of astronomical sources; for example, falsely identifying $z\sim3-4$ passive galaxies as $z>10$ galaxies leads to incorrect number counts and flawed tests of cosmology. In this work, we provide a novel but simple tool to better quantify the detection of faint lines in interferometric data sets and properly characterize the underlying noise distribution. We demonstrate the method on three sets of archival observations of $z>10$ galaxy candidates, taken with the Atacama Large Millimeter/Submillimeter Array (ALMA). By jackknifing the visibilities using our tool, $jackknify$, we create observation-specific noise realizations of the interferometric measurement set. We apply a line-finding algorithm to both the noise cubes and the real data and determine the likelihood that any given positive peak is a real signal by taking the ratio of the two sampled probability distributions. We show that the previously reported, tentative emission-line detections of these $z>10$ galaxy candidates are consistent with noise. We further expand upon the technique and demonstrate how to properly incorporate prior information on the redshift of the candidate from auxiliary data, such as from JWST. Our work highlights the need to achieve a significance of $\gtrsim 5\sigma$ to confirm an emission line when searching in broad 30 GHz bandwidths. Using our publicly available method enables the quantification of false detection likelihoods, which are crucial for accurately interpreting line detections.