The HTTP/2 draft -13 seems pretty final with respect to the Connection:
header being a HTTP/1.x thing only.

Move the handling accordingly
Be pendatically correct and Collect multiple Connection headers.
Unify the handling of the Connection: header between client/backend
The chunk-counter is http/1 specific
Make Connection: handling an entirely internal http1 issue.
Fix comments
Remove unused member of struct http_conn