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Rixasha: Mine is still saying "Looking for eligible games. Due to exceedingly high demand this may take up to several days, please check back later. Thank you for your patience." since more than a day.

Since it's only up for a couple of days I'm starting to think I'm going to miss this offer :(
Same issue here. I can see the games listed on the page when I am not logged in but as soon as I log in its as if there are no current games available.
If it isn't working for you, complete the following steps:

1) Contact GOG support and ask them to re-sync your Steam account to GOG
2) Run the game(s) you wish to sync for 2-3 minutes (launch from Steam), wait 20-30 minutes, and refresh GOG Connect
Post edited October 13, 2017 by oncologist
Shep7000: ...
Backlogger: ...
As time is drawing short I filed a support ticket this afternoon detailing my problem. I received a reply a bit over three hours later saying they forced a resync for me. And then it worked.