Can You Update Chromebooks After AUE
It would depend on what version you are currently using, if it's not the latest version for your device then it should get an update.
Older Chromebooks have older hardware parts, and these parts eventually lose the ability to get the latest updates. If your Chromebook is more than 5-8 years old depending on the model, you might see this message: "This device will no longer receive software updates. You can continue using your computer but should consider upgrading."
Please note that even though you have received that message it will likely be two years or more before certain websites may not load properly due to the Chrome browser being out of date. So it is nothing to be concerned about.
What you're at risk of is website access
Banks, Medical sites, and Streaming services all require more security than average sites so, those will fail first.
If none of the above are of concern then no issues short term.
Other Options
You could install Linux or
MrChromebox operating systems.
If this issue is happening on a new Chromebook you should return it and get a refund.
Hope this helped
Fred