With the spread of eyewear devices, people are increasingly using information devices in various everyday situations. In these situations, it is important for eyewear devices to have eye-based interaction functions for simple hands-free input at a low cost. This paper proposes a gaze movement recognition method for simple hands-free interaction that uses eyewear equipped with an infrared distance sensor. The proposed method measures eyelid skin movement using an infrared distance sensor inside the eyewear and applies machine learning to the time-series sensor data to recognize gaze movements (e.g., up, down, left, and right). We implemented a prototype system and conducted evaluations with gaze movements including factors such as movement directions at 45-degree intervals and the movement distance difference in the same direction. The results showed the feasibility of the proposed method. The proposed method recognized 5 to 20 types of gaze movements with an F-value of 0.96 to 1.0. In addition, the proposed method was available with a limited number of sensors, such as two or three, and robust against disturbance in some usage conditions (e.g., body vibration, facial expression change). This paper provides helpful findings for the design of gaze movement recognition methods for simple hands-free interaction using eyewear devices at a low cost.