The ScummVM version of BR had a port for Linux, which works fine on my system. I noticed that the Enhanced Edition was only available for Windows, so I contacted GOG to see what's up with that. Their reply:

"Hello,

Thank you for contacting us.

Blade Runner Enhanced edition is available for Windows operating systems only. We weren't able to make the game work reliably on our test computers running that system. Because of this, Linux operating system is not listed among the compatible ones on the product page. I apologize for the inconvenience.

Have a nice day.

Regards,
Orcan
GOG Customer Support"

On the bright side, it looks like I'll be saving some money... ;-)

Luckly, you can buy the previous version which supports Linux as far as I can tell.

https://www.gog.com/game/blade_runner

I do own the original already, I was just curious about the Enhanced Edition. The original is better, from all the reviews I've read, so I should be thankful! ;-)