The Eau Claire River is open from the Dells of the Eau Claire River County Park downtstream to Badger Road. Unsure about ice and navigability upstream from the park.
The USGS currently doesn't have streamflow data for the river; the river appears to be moving 3-5 knots.
The section from the county park to Badger Road is a very scenic paddle with three sections of rapids that are scoutable, and should be.
Consult "Whitewater Quietwater" by Bob Palzer for general information about this section. Although the most challenging rapid, which is within a mile downstream of the park, is not identified or described in the guide book.
This run was in a tandem canoe with a skirt; the boat would have taken large amounts of water without the skirt. Happy paddling!