UPDATE - with UMIT OZAYDIN we've investigated this trouble. The first step was to check the info from the error log (placed in logs/bx_pp_chargebee.log file. The latest messages from there had the following look:

12-09 18:56:47: Create Hosted Page Error: IO exception occurred when trying to connect to [test_checkout_url].chargebee.com/api/v2/hosted_pages/checkout_new . Reason : SSL: no alternative certificate subject name matches target host name '[test_checkout_url].chargebee.com'(0)

In the payment settings on UNA side was mentioned only [test_checkout_url] looking like sitename-test. So after the set the proper URL like [test_checkout_url].chargebee.com this error has gone away.

The second flaw was the wrong unique ID of permissions plans in Chargebee->Products Catalog->Plans which generated the error like:

12-11 00:33:02: Create Hosted Page Error: subscription[plan_id] : The pro-1-month referenced in parameter subscription[plan_id] is not present (0)

After the creation of a new plan with a Unique id like pro-1-month this error has passed too.