Thank you for all your help ralphy!
I really appreciate that you are doing this just because of my request.

My test results:

I have downloaded and scp'd the files to my Sqeezebox after connecting it via LAN.
Afterwards I stopped the WiFi service, made a backup of the original files and replaced them with the ones from the download.
Then I started the WiFi service again, which worked without any error.
I disconnected LAN and tried to connect to my old WiFi which is WPA2-PSK, which worked without any problems.

Until this point I can say: At least, it hasn't become worse

Now I reconnected LAN, transferred again all files and the configuration for my WPA2-EAP connection and tried to do the WPA2-EAP connection.
It didn't work, but the error message gave clues now, why.

After various different configurations of wpa_supplicant I finally found a config that successfully authenticated against my RADIUS server.

I replaced /etc/wpa_supplicant.conf with my successfull test-config and replaced /etc/network/interfaces as described in the tutorial. After rebooting, the device did not connect automatically to WiFi and did not find any WiFIs to connect to. Even LAN connections did not work any more and I had to factory reset. I guess, there was a problem with my /etc/network/interfaces file, but I have not the time to debug this now.

I'll give it another try later, so stay tuned. After everything works, I'll provide a little howTo for other interested people here.