Yes, possibly, and yes. I would test out the dairy thing first, as that is a common culprit. I have eczema, and I know that sometimes it itches and sometimes it hurts, and sometimes I don't feel it at all, so I can't say for sure if it's itchy for a baby. However, in areas such as behind my knees, where it's warm and moist, it hurts or itches more than other places, especially when it's hot out.
I avoid soap and anything with fragrance, only wear all cotton clothing, and avoid a lot of hot water. I don't know how old your baby is, but I also avoid fried foods, spicy foods, and chocolate as they seem to aggravate it.
There are a lot of awesome threads on here about eczema. Do a search with the word eczema and you will find a lot of info, from homemade remedies to diet solutions.