You should handle the TextChanged-Ebvent from both Textboxes.
That could be look like :
Private Sub TextBox_TextChanged(sender As Object, e As EventArgs) Handles ΟνομαTextBox.TextChanged , ΚωδικοςTextBox.TextChanged
If ΟνομαTextBox.Text = "" Or ΚωδικοςTextBox.Text = "" Then
Button2.Enabled = False
Else
Button2.Enabled = True
End If
End Sub