According to Swift docs https://docs.openstack.org/swift/latest/overview_expiring_objects.html adding X-Delete-After header to a PUT or POST object will make it expire after the seconds passed in that value.
Is this feature supported in IBM Bluemix Object Storage?
The IBM Bluemix Object Storage supports expiring objects, i.e., automatic deletion. You can use the X-Delete-At or X-Delete-After header attributes that you mentioned in your question. As usual, this is found in the documentation for IBM Bluemix Object Storage under "Managing Objects".
The X-Delete-At uses epoch time and this command (taken from the docs) would have deleted the object at "2016/04/01 08:00:00":
swift post -H "X-Delete-At:1459515600" container1 file7