The primary goal of this article is to extend the reciprocal LASSO for applications to binary and survival outcomes. We consider the least squares approximation (LSA) as a solver for the reciprocal LASSO problem. The LSA is a general theoretical framework that includes generalized linear models, Cox regression, and many others as special cases. By applying LSA to reciprocal LASSO regularization, we transfer the original reciprocal LASSO problem into an asymptotically equivalent least squares problem. While the existing literature on reciprocal LASSO has mostly focused on linear models, our algorithm can be easily implemented for general likelihoods, providing a flexible framework for variable selection using reciprocal penalties. To handle the computational burden of implementing the resulting procedure, we employ a scalable stochastic search method called Simplified Shotgun Stochastic Search with Screening (S5), which is easy to implement, without requiring any sophisticated optimization package other than a linear equation solver. We examine the effectiveness of our procedure through Monte Carlo simulations and real data analyses.