The term "degasifier" has been used to refer to two types of water treatment equipment: a vacuum degasifier and a "forced Draft degasifier" (also called a decarbonator).
Anyway, to answer your original question. A continuous flow of water through the unit will not result in freezing. The air flow is usually sized at 3 SCFM / GPM of water. Assuming inlet water temperature of 60 ^F, there will be some cooling, but not enough to result in freezing. Hopefully, the supplier of the degasifier provided a mist extractor at the top vent to prevent water carry-over. If not then you will see ice accumulating on the outside of the tank.