Alternative for Asus EEE PC that supports GNU/Linux out of the box?

I'd also suggest a used Thinkpad. Specifically the X series of Thinkpads if you want a laptop on the smaller EEE PC end of the spectrum.

I'd recommend looking at models from the X220 onward. The X220 though, is a firm favourite in the community for a number of reasons. It was the last X series Thinkpad to come with the old style (and almost universally said to be) superior keyboard. It was the first generation to have Sandy Bridge processors, which saw a sizeable jump in performance. And as it's one of the most popular models amongst enthusiasts, resources for modding, upgrading, fixing, and repairing them, be it in hardware or software, are plentiful.

Just bare in min that if you were to get an X220, you'd need the i7 CPU version in order to get USB 3.0.

The X220 is a favourite of Thinkpad fans but my recommendation would still hold true for later models such as the X230, X240 etc... There is one more thing to keep in mind though. Thinkpads are/were business laptops. They were sold by the thousands to big business. They were known for being durable and perhaps as a result of that reputation, a lot of them saw a tough life. So it's all too common to finnd ones for sale that are in poor condition. When buying a Thinkpad, pay close attention to the screen's hinges, the bezel around the screen, and the corners of the main body. If you wanted, you can always ask for further advice on the condition or price of a Thinkpad that's for sale, one one of the Thinkpad community's forums.

[NOTE] There was a EEE PC style series of Thinkpads but they were rubbish, not worthy of the Thinkpad name.