Upgradation of technologies for sustainable smart cities has led to rapid growth in Internet of Things (IoT) applications, including e-healthcare services wherein smart devices collect patient data and deliver it remotely to the servers in real-time. Despite its enormous benefits, IoT in healthcare has not received much attention primarily due to the risk of unauthorized access to confidential medical information enabled by the vulnerable wireless channel for communication. Besides, tiny IoT devices have limited computing power and storage capabilities that prevent administrators from using complex and resource-hungry security protocols. The cyber attacks on the Internet of Healthcare applications (IoHA) could result in fatalities, decreased revenue, and reputation loss, hence endangering sustainability. The existing security protocols are unsuitable due to the cost complexities that necessitate developing new security protocols for resource-constrained and heterogeneous IoT networks. We introduce a confidentiality and anonymity-preserving scheme for critical infrastructures of IoT to conquer cyber threats for sustainable healthcare. This paper proposes Zero-Knowledge Proofs (ZKP) based Authenticated Key Agreement (AKA) protocol for IoHA. ZKP-AKA uses zero-knowledge proofs, physically unclonable function, biometrics, symmetric cryptography, message digest, etc., for accomplishing the protocol's objective at minimal computation, storage, and communication expenses. ZKP-AKA retains data integrity, confidentiality, anonymity, and safety from significant cyber threats.