Maybe the following will help...
I've noticed that Netflix only logs me out after trying to play back any contents (after entering the actual player).
However if I just hover over the thumbnail and NOT click on it, wait for a second for a 'zoom-in' and then click on a down-arrow below the thumbnail on the right, the Netflix info window will pop out, SF will properly analyze the contents and Netflix will NOT log me out.
If in any case Netflix logs you out - logging back in should be preceded by clearing the cookies, like in the post #7 above.