In your first example, you are creating a String, "silly" and then passing it as a parameter to another String's copy constructor, which makes a second String which is identical to the first. Since Java Strings are immutable (something that frequently stings people who are used to C strings), this is a needless waste of resources. You should instead use the second example because it skips several needless steps.
However, the String literal is not a CaseInsensitiveString so there you cannot do what you want in your last example. Furthermore, there is no way to overload a casting operator like you can in C++ so there is literally no way to do what you want. You must instead pass it in as a parameter to your class's constructor. Of course, I'd probably just use String.toLowerCase() and be done with it.
Also, your CaseInsensitiveString should implement the CharSequence interface as well as probably the Serializable and Comparable interfaces. Of course, if you implement Comparable, you should override equals() and hashCode() as well.