Known MembersAmaterasu.Haumea.Inca Kasugatani.Iris.Nataku Son.Shinra Kusakabe.Shō Kusakabe.Sumire.
Nope, he doesn't. If you have seen the borutos movie then there you could have seen that when momoshiki uses byakugan to see who is nine tail jinchuuriki, you could have seen that boruto had just some normal chakra and no kyuubi chakra.
Naruto: Momoshiki Is Basically Boruto's Nine-Tails.
No, Boruto never gets Byakugan but he doesn't need Byakugan he already had jougan(pure eye). It's is far stronger than Byakugan and only otsutsuki clan possess it. Even it's clan members are referred jougan as a touble some. So, he doesn't need Byakugan.