Yom Kippur is definitely not a sad day. We are sad and mournful on Tish’a b’Av, when we remember the destruction of both Temples and other terrible things that have happened to our people. Yom Kippur is a solemn and important day, but it is not sad. We fast in order to focus and be purified, not as a sign of mourning.
Some rabbis have referred to Yom Kippur as the happiest day of the Jewish year, because it is our opportunity to make things right with G-d and our fellow humans. Many people certainly feel peaceful and relieved at the end of the fast, knowing that they have laid their sins at G-d’s feet and been forgiven.
Still, it is not appropriate to wish people a Happy Yom Kippur. The proper greeting is “G’mar chatimah tovah” – a good final sealing [in the Book of Life].
One can also wish another an easy fast.
Sources
Is Yom Kippur a Happy or a Sad Day?
Judaism 101: Expressions and Greetings
Sad or happy on Yom Kippur?
