Hey guys, I totally agree with Eliza about encouraging them to potentially book a room to practise and even more importantly- being silly during the lessons. I too have a couple of students who lack confidence and I have taken the approach of distraction, which seems to have worked pretty well thus far.
During our vocal warm ups, I ask them to perform a physical exercise (gentle head shake from side to side, or lifting up shoulders and letting them fall just before doing a lip trill, etc). What I have found is that they then shift their focus from thinking about ‘producing sound / singing’ to performing the physical movement, which in turn results in the vocal exercise being performed in a much more relaxed manner.