Why are they allowed to call it the "Eid holiday", but have to use "Winter break" and "Spring break" instead of Christmas and Easter?
"Eid holiday" is redundant. "Eid" just means holiday.
Probably because we haven't thought of a good euphemism yet. Christmas and Easter fall near the Winter and Spring Solstices. In their honor, schools typically give not just a day off, but a week or two weeks-- hence there's a real "break."
Eid is just one day, so there's no "break" that can be structured around it-- similarly, Rosh H'Shana and Yom Kippur are just 3 days collectively. There's no p.c. way to hide the reason for taking off school-- unless you wanted to honor those days by giving students a full week off, and calling it "Fall Break" or something.