Thanks Mario.

I found the solution:

The problem was the differents between http and https. My Odoo Server has a https certificate. My POSBOX not. So the browser don't allow the communication to the POSBOX. So I installed a certificate on POSBOX and set the IP on Odoo hardware proxy to "https://x.x.x.x

Now it works ;)

So, if Odoo runs in the cloud, I would need a second PC anyhow?  Then I guess, it would be easier connect printers etc. directly to that pc, until I would need another point of sales, right? Are there any hardware requirements for PC? I guess any device that runs a webbrowser and has an ethernet adaptor should do, the job, right?

The POSbox needs to be in the same network as the PC, where you run the Point of Sale. When you start the POS screen, it wil search for the POSbox. Look at the network icon right above the screen. I've tested it on Firefox and Chrome with Odoo SaaS and my own Odoo VPS.