Eunice and her husband Richard were likely first cousins. I am not certain that Eunice is a child of Joseph & Amanda (Sackett) Remington. However, her maiden name is Remington, and she is buried in the same cemetery as Joseph & Amanda and other family members. Based on her naming of a daughter Lydia Amanda, I feel she does belong as a daughter of Joseph & Amanda, sho also had a daughter named Lydia A. Eunice's mother-in-law is Melinda Sackett, sister of her (assumed) mother.
One webpage has Eunice's birthdate as 20 Apr 1831 (Descendants of John Sackett).