Although it sounds like she has a French given name, she and her sister were named after the daughters of Prince Andrew and Sarah Ferguson:
Eugenie and Beatrice. Her mother (who is an anglophone of Irish descent) says that she prefers the English pronunciation of her name and likes for it to be spelled without an accent.
In the interview you linked to, her dad has the accent of a typical Anglo Montrealer--either he's completely bilingual or he speaks English better than French. He's owns a financial consulting firm in Westmount.
Eugenie, on the other hand, seems to have picked up a slight American accent, which would make sense since she's spent most of her life in Florida.
EB: Westmount High School has classes for students from grades 7 to 11 (Sécondaire 1-5, as we say here), so it's quite possible she only attended the school when she was a young teenager.