Will I always need to keep an everstone on my Eevee to keep it as one?
(I love my shiny boy).
Only if you use a rare candy. As of Sword and Shield, you can evolve level 100 Pokemon (that evolve via levelling up) by using a rare candy. As you need to level up your eevee with max friendship to evolve it, it should be fine. Even if you do use a rare candy on it, you can cancel the evolution.