boolean compare(String str1, String str2) {
    if(str1==null || str2==null) {
        //return false; if you assume null no equal to null
        return str1==str2;
    }
    return str1.equals(str2);
}

is this what you desired?

boolean compare(String str1, String str2) {
    if(str1==null || str2==null) {
        //return false; if you assume null not equal to null
        return str1==str2;
    }
    return str1.equals(str2);
}

is this what you desired?

boolean compare(String str1, String str2) {
    if(str1==null || str2==null) {
        return false;
    }
    return str1.equals(str2);
}

is this what you desired?

boolean compare(String str1, String str2) {
    if(str1==null || str2==null) {
        //return false; if you assume null no equal to null
        return str1==str2;
    }
    return str1.equals(str2);
}

is this what you desired?