I think both names are very nice on their own, but not for twins. Twins are two individual people who won't be sharing a womb forever, and they should be treated as such, even in the names. My litmus test for twin names is if you would use them separately for an individual boy and girl then fine; however, if you would only use
June and
Jesse in the case of boy/girl twins then it bothers me because you are picking names for theme reasons which isn't my personal style.
That said,
June is charming and sweet and
Jesse is strong and handsome, but I don't like the matching initial for twins.