If you want to check if a string is empty:

if (str[0] == '\0')
{
    // your code here
}

You can leverage Apache Commons StringUtils.isEmpty(str), which checks for empty strings and handles null gracefully.

Example:

System.out.println(StringUtils.isEmpty("")); // true
System.out.println(StringUtils.isEmpty(null)); // true

Google Guava also provides a similar, probably easier-to-read method: Strings.isNullOrEmpty(str).

Example:

System.out.println(Strings.isNullOrEmpty("")); // true
System.out.println(Strings.isNullOrEmpty(null)); // true

String.EMPTY is 12 characters, and "" is two, and they would both be referencing exactly the same instance in memory at runtime. I'm not entirely sure why String.EMPTY would save on compile time, in fact I think it would be the latter.

Especially considering Strings are immutable, it's not like you can first get an empty String, and perform some operations on it - best to use a StringBuilder (or StringBuffer if you want to be thread-safe) and turn that into a String.

Update
From your comment to the question:

What inspired this is actually TextBox.setText("");

I believe it would be totally legitimate to provide a constant in your appropriate class:

private static final String EMPTY_STRING = "";

And then reference it as in your code as

TextBox.setText(EMPTY_STRING);

As this way at least you are explicit that you want an empty String, rather than you forgot to fill in the String in your IDE or something similar.

To detect if a string is null or empty, you can use the following without including any external dependencies on your project and still keeping your code simple/clean:

if(myString==null || myString.isEmpty()){
    //do something
}

or if blank spaces need to be detected as well:

if(myString==null || myString.trim().isEmpty()){
    //do something
}

you could easily wrap these into utility methods to be more concise since these are very common checks to make:

public final class StringUtils{

    private StringUtils() { }   

    public static bool isNullOrEmpty(string s){
        if(s==null || s.isEmpty()){
            return true;
        }
        return false;
    }

    public static bool isNullOrWhiteSpace(string s){
        if(s==null || s.trim().isEmpty()){
            return true;
        }
        return false;
    }
}

and then call these methods via:

if(StringUtils.isNullOrEmpty(myString)){...}

and

if(StringUtils.isNullOrWhiteSpace(myString)){...}

Use org.apache.commons.lang.StringUtils

I like to use Apache commons-lang for these kinds of things, and especially the StringUtils utility class:

import org.apache.commons.lang.StringUtils;

if (StringUtils.isNotBlank(str)) {
    ...
} 

if (StringUtils.isBlank(str)) {
    ...
}