No. These are not pickled eggs. These are homegrown, unwashed eggs stored in lime water

The lime water fills all the pores of the eggs and encases them in a “glass” shell. Water-glazed eggs can last stored at room temperature this way for up to 2 years. This method of preserving raw eggs has been used since the 1800s and was common even in the 1940s and 1950s. When refrigerators became a standard kitchen appliance, water-glazing almost became a lost art.
You cannot use commercial eggs for this because they have all had the protective layer (bloom) removed from the shell and will spoil quickly. I recently scrambled 18 eggs that had been stored in lime water for 7 months on a shelf in an unrefrigerated cupboard and they tasted perfectly fresh (although the yolk seemed a bit thinner than fresh eggs).