Twitter data has been shown broadly applicable for public health surveillance. Previous public health studies based on Twitter data have largely relied on keyword-matching or topic models for clustering relevant tweets. However, both methods suffer from the short-length of texts and unpredictable noise that naturally occurs in user-generated contexts. In response, we introduce a deep learning approach that uses hashtags as a form of supervision and learns tweet embeddings for extracting informative textual features. In this case study, we address the specific task of estimating state-level obesity from dietary-related textual features. Our approach yields an estimation that strongly correlates the textual features to government data and outperforms the keyword-matching baseline. The results also demonstrate the potential of discovering risk factors using the textual features. This method is general-purpose and can be applied to a wide range of Twitter-based public health studies.