In java there is nothing as empty character literal, in other words, '' has no meaning unlike "" which means a empty String literal
The closest you can go about representing empty character literal is through zero length char[], something like:
char[] cArr = {}; // cArr is a zero length array
char[] cArr = new char[0] // this does the same
If you refer to String class its default constructor creates a empty character sequence using new char[0]
Also, using Character.MIN_VALUE is not correct because it is not really empty character rather smallest value of type character.
I also don't like Character c = null;
as a solution mainly because jvm will throw NPE if it tries to un-box it. Secondly, null is basically a reference to nothing w.r.t reference type and here we are dealing with primitive type which don't accept null as a possible value.
Assuming that in the string, say str, OP wants to replace all occurrences of a character, say 'x', with empty character '', then try using:
str.replace("x", "");