Do eggs contain all my needed amino acids?

YES. Eggs are such a good and complete source of amino acids/protein that they are the standard to which other foods are compared for adequacy of protein. It is wise to choose organic eggs and ones that are pasture-raised if possible. See http://www.Whfoods.Com/genpage.Php?Tname=foodspice;dbid=92 for a good discussion of the nutritional content of eggs.