Is this sentence correct?
May you stay forever young!

Yes, it's correct. It's also the title of a Bob Dylan song:

"Forever Young"

May God bless and keep you always
May your wishes all come true
May you always do for others
And let others do for you
May you build a ladder to the stars
And climb on every rung
May you stay forever young
Forever young, forever young
May you stay forever young.

May you grow up to be righteous
May you grow up to be true
May you always know the truth
And see the lights surrounding you
May you always be courageous
Stand upright and be strong
May you stay forever young
Forever young, forever young
May you stay forever young.

May your hands always be busy
May your feet always be swift
May you have a strong foundation
When the winds of changes shift
May your heart always be joyful
And may your song always be sung
May you stay forever young
Forever young, forever young
May you stay forever young.

Yes, "forever" is an adverb that can be used before or after the modified verb or adjective.  When used before, is has a bit more formal or classic sound, at least to my ear.  When used after the modified word, the meaning tends toward "without ever ending."  When used before, it also caries a sense of "continually."

It sounds right. I think it's part of a TV commercial for diamonds.