Got a mosquito bite last summer around my eye. Now, everytime I go out in the cold, it turns red like I got a bite again. Why is this happening?

Residual allergy. The is hypersensitivity to walled off foreign material. Cold is causing the area to react and swell, creams coukd help but if it very localized a microscopic foreign body may be persitent. See opthamologist if minor surgery is an option.
Reaction. Sometimes after an insect bite the body develops a persistent bite reaction. This reaction can come and go as you describe. There are also other similar conditions that would really need a dermatologist's eye.