Data security and user privacy are two main security concerns in the cold-chain logistics system (CCLS). Many security issues exist in traditional CCLS, destroying data security and user privacy. The digital signature can provide data verification and identity authentication based on the mathematical difficulty problem for logistics data sharing in CCLS. This paper first established a blockchain-enabled cold-chain logistics system (BCCLS) based on union blockchain technology, which can provide secure data sharing among different logistics nodes and guarantee logistics data security with the untampered blockchain ledger. Meanwhile, a lattice-based undeniable signature scheme is designed to strengthen the security of logistics data and user privacy against quantum attacks. This scheme is based on the lattice assumption, which can provide anti-quantum security for BCCLS in the future quantum computer age. It also establishes the undeniable mechanism that guarantees that the actual signer cannot deny a valid signature. Then, security proof shows that this undeniable signature scheme is correct and safe, and an efficiency comparison shows that it is more efficient than other schemes in similar literature. The performance evaluations of transaction throughput and latency show that the proposed BCCLS is efficient and practical. This work can improve the security and efficiency of logistics data management and promote the reform and development of the logistics industry.