What do the white crescents and specks on peoples nails mean, is it a warning sign for health problems?

No. White specks on nails or leukonychia punctata is a harmless condition and thought to represent the after effects of minor injury to the base of the nail. The spots generally disappear on their own within a few days. It is not a sign of a lack of vitamin or calcium in the diet or health problems.
White specks. If it is on all the nails then sometimes it is a nutritional deficiency. Mineral deficiencies especially zinc deficiency can show up as white specks on the nails. Try adding calcium'/magnesium/zinc to your vitamins. Get zinc in other forms rather than zinc oxide which is not absorbed well.