No. This is not possible since border policies were updated following 2001 terrorist attacks. The St. Croix River is an international border.
There are ample sites on both sides of the river, and portages on each side, that allow you to make your trip while going to shore only in one country. It is important that you launch, go to shore, camp, portage and take out in the same country only. If you go to shore, you are entering another country and are responsible for reporting to the appropriate Customs authority; failure to do so may result in arrest. The riverbanks and river are patrolled by Customs agents. See our Trip Planning section for information.