There are NEVER ice fishing shacks on the river. In both Lakes I would say there probably are, as sturgeon spearing season in Winnebago starts in about a month. From what I recall (I worked in Oshkosh 20 yrs or so ago), the river never seems to freeze completely over due to the current.
The river is a stretch a couple miles long between the two lakes (Winnebago and BTM) and maybe a 100-150 yards wide in most places.