Randomize text into random boxes

Let me see if I understand you first - your explanation isn't that clear.
You have four text boxes:

  1       2      3      4
Hello   There   My   Friend

And you want to press a button and "swap" the text about randomly:

  1       2      3      4
There  Friend   My    Hello

Then

  1       2      3      4
My     Friend  There  Hello

And so on.

That's not difficult to do, but to make it easier, create an array of strings and copy the text into it:

Dim data As String() = New String(3) {}
data(0) = textBox1.Text
data(1) = textBox2.Text
data(2) = textBox3.Text
data(3) = textBox4.Text

Now, you can just use the index number to decide what to move around.
So create a Random instance, and run a loop:

Dim rand As Random = New Random()

For i As Integer = 0 To 10 - 1
    ...
Next

The loop runs more times than there are elements to really mix them about!
Inside the loop, generate two indexes from the Random generator, and swap the elements:

Dim index1 As Integer = rand.[Next](4)
Dim index2 As Integer = rand.[Next](4)
Dim temp As String = data(index1)
data(index1) = data(index2)
data(index2) = temp

Now all you have to do is put the strings back into the text boxes!

textBox1.Text = data(0)
textBox2.Text = data(1)
textBox3.Text = data(2)
textBox4.Text = data(3)

All done!