The healthcare sector is now blending with Information and Communications Technology (ICT) using Internet of Things (IoT) to potentially minimise medical errors and reduce healthcare cost. Patients are now embedded with smart devices like body sensors and wearable devices which can monitor their health without the need for a doctor in physical contact. Such smart devices have the downside of low battery power and are unable to transmit their data to the medical personnel when the patient is on the move away from the smart home/smart clinic fixed gateway. A mobile gateway is required which moves with the patient to process the smart device data without depleting the smartphone battery. This paper proposes an Intelligent and Energy Efficient SG based smartphone Gateway for healthcare smart devices (IEE5GG). In IEE5GG, the 5G architecture is adopted and the patient's smartphone is used as a gateway where multiple smart devices are connected e.g. via Bluetooth. To save energy, requests to the smartphone can either be executed on the smartphone gateway or offloaded and executed in the Mobile Edge Computing (MEC) cloud at close proximity to the smartphone in the 5G Base Station (BS) central Unit (gNB-CU) while considering the transmission power, Quality of Service (QoS), smartphone battery level and Central Processing Unit (CPU) load. Results show that the proposed IEE5GG framework saves up to 38% of energy in the healthcare mobile gateway smartphone and reduces healthcare application service time by up to 41%.