We were flying from the airport and the flight was over night leaving Paris late at night and arriving in Washington the next morning.
Can we use "over day" instead of "over night" (or "during the day"?)
If I were a native speaker of English, I would never shut up. :-)
No. It is "overnight," one word, and there is no "daytime" equivalent. Things are presumed to happen during normal daytime or evening hours. The fact that something takes place "overnight" is the exception that needs to be mentioned.
These types of flights are known as "redeyes."