Don't know if you are still having problems but the advice on Netflix for this error with just about everything is to restart your device, sign out and then sign in again. You can restart your 360 by turning the power off and then back on with the switch on the back or at the mains socket, leave off for about a minute. If you can't get to the sign out stage then do a factory reset on the 360 you will then have to sign in again.
You have the option to keep you recordings or format the HDD, you can keep recordings and series links it's not a real factory reset that deletes everything, you will have to reenter Netflix login details and any other app logins you have set up.
Not a user of Netflix but are there limitations on how many devices can be logged in at once. Maybe logout on another device and see if you still have the same problem.
I am more than happy to take a look at this for you
I will need to send you a private message to pass security
If you can check the purple envelope top right of your screen that would be great