Modern engineering increasingly relies on vast datasets generated by experiments and simulations, driving a growing demand for efficient, reliable, and broadly applicable modeling strategies. There is also heightened interest in developing data-driven approaches, particularly neural network models, for effective prediction and analysis of scientific datasets. Traditional data-driven methods frequently involve extensive manual intervention, limiting their ability to scale effectively and generalize to diverse applications. In this study, we propose an innovative pipeline utilizing Large Language Model (LLM) agents to automate data-driven modeling and analysis, with a particular emphasis on regression tasks. We evaluate two LLM-agent frameworks: a multi-agent system featuring specialized collaborative agents, and a single-agent system based on the Reasoning and Acting (ReAct) paradigm. Both frameworks autonomously handle data preprocessing, neural network development, training, hyperparameter optimization, and uncertainty quantification (UQ). We validate our approach using a critical heat flux (CHF) prediction benchmark, involving approximately 25,000 experimental data points from the OECD/NEA benchmark dataset. Results indicate that our LLM-agent-developed model surpasses traditional CHF lookup tables and delivers predictive accuracy and UQ on par with state-of-the-art Bayesian optimized deep neural network models developed by human experts. These outcomes underscore the significant potential of LLM-based agents to automate complex engineering modeling tasks, greatly reducing human workload while meeting or exceeding existing standards of predictive performance.