Why are my eye whites greying?

Really greying?: Sometimes melanin pigment deposits can give appearance of graying. This is normal. In older age grey plaque in sclera are normal. If your sclera is progressively getting grey it may be thinning due to "collagen vascular diseases', although rare but since you are only 34 you must see an ophthalmologist that then will refer you to a rheumatologist for more workup & diagnosis if necessary.

Depends: If this is not a sign of systemic disease then it is heredity.