Asymmetric information stochastic games (AISGs) arise in many complex socio-technical systems, such as cyberphysical systems and IT infrastructures. Existing computational methods for AISGs are primarily offline and can not adapt to equilibrium deviations. Further, current methods are limited to particular information structures to avoid belief hierarchies. Considering these limitations, we propose conjectural online learning (COL), an online learning method under generic information structures in AISGs. COL uses a forecaster-actorcritic (FAC) architecture, where subjective forecasts are used to conjecture the opponents' strategies within a lookahead horizon, and Bayesian learning is used to calibrate the conjectures. To adapt strategies to nonstationary environments based on information feedback, COL uses online rollout with cost function approximation (actor-critic). We prove that the conjectures produced by COL are asymptotically consistent with the information feedback in the sense of a relaxed Bayesian consistency. We also prove that the empirical strategy profile induced by COL converges to the Berk-Nash equilibrium, a solution concept characterizing rationality under subjectivity. Experimental results from an intrusion response use case demonstrate COL's faster convergence over state-of-the-art reinforcement learning methods against nonstationary attacks.