I already exposed to Philips HUE bridge some non Philips HUE led strips using controllers like this one.
I am wondering: maybe I can integrate the Home Connect light into Philips HUE bridge using Home Assistant and a Zigbee antenna.
What I want to achieve:
Using Philips HUE remote to turn on/off chicken lights is triggering Home Connect cook hood lights on/off as well.
Not sure what you mean by “zigbee antenna”, but, basically, what you ask is not possible.
What many of us did is to use a 3rd party zigbee adapter connected to HA and link the Hue devices (lights/remotes/…) to that “coordinator” rather than the bridge, which then become superfluous.
What you can also do is use the Hue integration to capture the remote events.