Because the value side is surrounded by double quotes, you can't use a double quote inside the value. however you can use the unicode equivalent of a double quote \x22
inside the value.
So your ValidationExpression attribute value would look something like:
ValidationExpression="^((?>[a-zA-Z\d!#$%&'*+\-/=?^_`{|}~]+\x20*|\x22((?=[\x01-\x7f])[^\x22\\]|\\[\x01-\x7f])*\x22\x20*)*(?<angle><))?((?!\.)(?>\.?[a-zA-Z\d!#$%&'*+\-/=?^_`{|}~]+)+|\x22((?=[\x01-\x7f])[^\x22\\]|\\[\x01-\x7f])*\x22)@(((?!-)[a-zA-Z\d\-]+(?<!-)\.)+[a-zA-Z]{2,}|\[(((?(?<!\[)\.)(25[0-5]|2[0-4]\d|[01]?\d?\d)){4}|[a-zA-Z\d\-]*[a-zA-Z\d]:((?=[\x01-\x7f])[^\\\[\]]|\\[\x01-\x7f])+)\])(?(angle)>)$"