Right mouse Button doesn't change the selection in the CheckListBox. Only the Left mouse button does. Please have a look at the sample CheckLists from the LiveExplorer app of the toolkit available here : https://wpftoolkit.codeplex.com/.
For the contextMenu, maybe you can override the "OnMouseRightButtonDown()" method from ChecListBox.
For the contextMenu, maybe you can override the "OnMouseRightButtonDown()" method from ChecListBox.