The all wood footboard of one of our beds has been squeaking at the
joint where the horizontal board attaches to the vertical post. I
tried spraying some silicone in there a few weeks ago to silence it but
to no avail.

Yesterday when my wife was changing the sheets somehow the board
separated from the post. Nothing is broken, it just came out of the
notches where the dowels and main load bearing connection are. I
pushed everything back together for now, but I doubt it will hold
together forever.

Why is there no glue in these connections? Everything seems to be dry
fitted together. Should I put some wood glue in there and hold it all
together with bungee cords until it dries?