This work is the first to take advantage of recurrent neural networks to predict influenza-like illness (ILI) dynamics from various linguistic signals extracted from social media data. Unlike other approaches that rely on timeseries analysis of historical ILI data and the state-of-the-art machine learning models, we build and evaluate the predictive power of neural network architectures based on Long Short Term Memory (LSTMs) units capable of nowcasting (predicting in "real-time") and forecasting (predicting the future) ILI dynamics in the 2011 - 2014 influenza seasons. To build our models we integrate information people post in social media e.g., topics, embeddings, word ngrams, stylistic patterns, and communication behavior using hashtags and mentions. We then quantitatively evaluate the predictive power of different social media signals and contrast the performance of the-state-of-the-art regression models with neural networks using a diverse set of evaluation metrics. Finally, we combine ILI and social media signals to build a joint neural network model for ILI dynamics prediction. Unlike the majority of the existing work, we specifically focus on developing models for local rather than national ILI surveillance, specifically for military rather than general populations in 26 U.S. and six international locations., and analyze how model performance depends on the amount of social media data available per location. Our approach demonstrates several advantages: (a) Neural network architectures that rely on LSTM units trained on social media data yield the best performance compared to previously used regression models. (b) Previously under-explored language and communication behavior features are more predictive of ILI dynamics than stylistic and topic signals expressed in social media. (c) Neural network models learned exclusively from social media signals yield comparable or better performance to the models learned from ILI historical data, thus, signals from social media can be potentially used to accurately forecast ILI dynamics for the regions where ILI historical data is not available. (d) Neural network models learned from combined ILI and social media signals significantly outperform models that rely solely on ILI historical data, which adds to a great potential of alternative public sources for ILI dynamics prediction. (e) Location-specific models outperform previously used location-independent models e.g., U.S. only. (f) Prediction results significantly vary across geolocations depending on the amount of social media data available and ILI activity patterns. (g) Model performance improves with more tweets available per geo-location e.g., the error gets lower and the Pearson score gets higher for locations with more tweets.