Sequential recommenders have been widely used in industry due to their strength in modeling user preferences. While these models excel at learning a user's positive interests, less attention has been paid to learning from negative user feedback. Negative user feedback is an important lever of user control, and comes with an expectation that recommenders should respond quickly and reduce similar recommendations to the user. However, negative feedback signals are often ignored in the training objective of sequential retrieval models, which primarily aim at predicting positive user interactions. In this work, we incorporate explicit and implicit negative user feedback into the training objective of sequential recommenders in the retrieval stage using a "not-to-recommend" loss function that optimizes for the log-likelihood of not recommending items with negative feedback. We demonstrate the effectiveness of this approach using live experiments on a large-scale industrial recommender system. Furthermore, we address a challenge in measuring recommender responsiveness to negative feedback by developing a counterfactual simulation framework to compare recommender responses between different user actions, showing improved responsiveness from the modeling change.