Is there a way to sign out of the Directv Stream box without it automatically signing back in?

You can try to call customer support to have them guide you through or otherwise manually turn off auto-sign in. However, DIRECTV STREAM boxes are generally locked to your account and cannot be reused (they are bricks once you cancel). 

As a note, your relative can use a streaming device such as a Fire Stick, Roku, or Apple TV for DIRECTV STREAM service, as a DIRECTV stream box is NOT required to receive service. The only benefits of a directv stream box are a better remote with a physical Guide button and channel number keypad, as well as faster speeds. 

Yes:

1. Go to your account overview.
2. In My streaming devices' area, select manage.
3. In the auto sign-in, change the selector from Off to On.
4. Follow the prompts to complete.

It's my understanding that once you disable it then you can not turn it back on.

I guess I'll have to call and see if they can do it. They don't have any other devices and they are not going to buy one just to see if they like it. They will if they like it. They've seen it at my house, but not actually use it in their home. I dont have another device they can use to try it. I wouldn't say the devices cant be reused because my newest was a used device. I had no issues reusing it, but it also wasnt signed into someone elses account.

bcbsncjlj as I said in my comment there are no devices listed in my account and no way to change it as there is no selector. I do remember way back turning it off which is probably why its not showing the option, however the devices still sign in automatically.

Devices are only listed in your account if they were purchased from DirecTV. If you purchased from a 3rd party like eBay or Amazon, etc. then they are not registered directly to you.

I have 1 device that was purchased from DirecTV. It does not show. 

You need to contact them and get it registered to your account.