You are trying to use an enum in a way it has not been made for; each element of the enum has to have an unique, constant value.
What you need is a function returning a value of your enum according to the input. Something like:
enum StringCompare { strA_is_less_than_strB, strAequals_strB, strA_is_greater_than_strB }
public StringCompare GetStringCompare(string lhs, string rhs) {
return
(lhs < rhs) ? StringCompare.strA_is_less_than_strB :
(lhs > rhs) ? StringCompare.strA_is_greater_than_strB :
StringCompare.strAequals_strB;
}