Is it possible to define an std::thread and initialize it later?

How about wrapping it in a pointer?

std::unique_ptr<std::thread> thread_ptr;

// Look into std::make_unique if possible
thread_ptr = std::unique_ptr<std::thread>(new std::thread(...));

Edit: And yes, the others have mentioned it and I didn't feel the need to add it here, but in order to avoid more downvote piling, I'll say it: You are passing *this and not this thereby copying an instance of your class. (Problems arise because it's non-copyable. Pass this and you should be good to go.)

Your problem is something else - you're passing an instance of MyClass into the thread instead of the pointer to MyClass which the member functions expect. Simply change DoDiskJobThread() like this (do not dereference this):

void MyClass::DoDiskJobThread()
{
    std::wstring Source = GetSource();
    std::wstring Target = GetTarget();
    int m_OperationType = GetOperationType();
    if      (m_OperationType == OPERATION_COPY)
    {
        DiskJobThread = std::thread(&MyClass::CopyThread, this, Source, Target);
    }
    else if (m_OperationType == OPERATION_MOVE)
    {
        DiskJobThread = std::thread(&MyClass::MoveThread, this, Source, Target);
    }
}

You were getting the error because *this resulted in trying to copy MyClass into the thread function, and the copy ctor of your class is deleted (because that of std::thread is deleted). However, the member functions CopyThread and MoveThread require a pointer as the first (hidden) argument anyway.

Live demonstration

You can't initialize the thread object after it's created; by definition, initialization occurs when an object is created. But you can use swap to move a thread object into another:

std::thread thr1; // no thread of execution
std::thread thr2(my_function_object); // creates thread of execution
thr1.swap(thr2);  // thr1 is now running the thread created as thr2
                  // and thr2 has no thread of execution