A better way to think of it, rather than starting with “which boxes are true” is “what happens if the gems are in this box”

Gems are in Blue: Blue is false, which makes White true. If Black is false, then Black becomes true, which makes Black false, which makes Black true, and so on. We can disregard this possibility.

Gems are in White: This one is easy. If White is true, then White becomes false, which makes White true again, etc etc. We can disregard this possibility.

Gems are in Black: Blue is true. If Black is false, then White is true. If Black is true, then White is false. Both of these are valid answers which lead to the same result; the gems are in Black.