Unit tests (UTs) play an instrumental role in assessing code correctness as well as providing feedback to large language models (LLMs), motivating automated test generation. However, we uncover a trade-off between generating unit test inputs that reveal errors when given a faulty code and correctly predicting the unit test output without access to the gold solution. To address this trade-off, we propose UTGen, which teaches LLMs to generate unit test inputs that reveal errors along with their correct expected outputs based on task descriptions. Since model-generated tests can provide noisy signals (e.g., from incorrectly predicted outputs), we propose UTDebug that (i) scales UTGen via test-time compute to improve UT output prediction, and (ii) validates and backtracks edits based on multiple generated UTs to avoid overfitting, and helps LLMs debug effectively. We show that UTGen outperforms other LLM-based baselines by 7.59% based on a metric measuring the presence of both error-revealing UT inputs and correct UT outputs. When used with UTDebug, we find that feedback from UTGen's unit tests improves pass@1 accuracy of Qwen2.5 32B on HumanEvalFix and our own harder debugging split of MBPP+ by over 3.17% and 12.35% (respectively) over other LLM-based UT generation baselines. Moreover, we observe that feedback from Qwen2.5 32B-based UTGen model can enhance debugging with frontier LLMs like GPT-4o by 13.8%. Lastly, we demonstrate that UTGen is a better judge for code correctness, outperforming a state-of-the-art trained 8B reward model by 4.43% on HumanEval+ with best-of-10 sampling using Qwen2.5 7B.