Spin squeezing and quantum entanglement are very important and widely used in quantum information processing. Therefore, it is meaningful to produce spin squeezing and quantum entanglement using the multi-body interaction in Bose-Einstein condensation. The effects of three-body and four-body interactions on spin squeezing and quantum entanglement at high density Bose-Einstein condensation are investigated, and the spin squeezing parameters and two entanglement parameters are analytically calculated using short time approximation. The results show that in the dynamic process, three- and four-body interactions can produce spin squeezing and quantum entanglement. Moreover, four-body interaction can produce stronger spin squeezing and better entanglement than the three-body interaction.