This paper investigates the entropy squeezing of a moving two-level atom interacting with the two-mode entangled coherent field via two-photon transition by using an entropic uncertainty relation and the degree of entanglement between the two-mode fields by using quantum relative entropy. The results obtained from numerical calculation indicate that the squeezed period, the duration of entropy squeezing and the maximal squeezing can be controlled by appropriately choosing the intensity of the light field, the atomic motion and the field-mode structure. The atomic motion leads to the periodic recovery of the initial maximal degree of entanglement between the two-mode fields. Moreover, there exists a corresponding relation between the time evolution properties of the atomic entropy squeezing and those of the entanglement between the two-mode fields.