Yes. Mount Eagle Elementary School is in Fairfax County.
Homes with zip codes 22303, 22306, 22307, 22308, 22309, 22310, 22312, and 22315 will have an Alexandria postal address, but are all located in Fairfax County. Students living in homes in those zip code areas will be attending a Fairfax County School.
At the end of World War II, there were about 50,000 people living in Fairfax County, and it was largely a farming area. Unincorporated areas outside of incorporated areas like Alexandria and Falls Church received postal addresses for the nearest incorporated area. This practice continues to this day and is often the source of confusion.
And for the benefit of the school teacher/realtor whose response included "your in Fairfax County," I would note that the correct word is "you're" not "your." "Your" is the possessive. As the first word in the sentence, it should have been capitalized.