Our practice is an odd thing to describe, but I'll try.
First, you can't 'stop' thinking. Trying to stop thinking is just more thinking.
Secondly, there really is no 'progress' to speak of. The whole idea of 'progress' is anathema to Shikantaza. There is 'effortless effort' - but that is an 'in each moment thing'. You will not get 'better' at Shikantaza because it's hard to qualify or quantify 'better'. Is having fewer thoughts an indicator of 'progress'? Not really. I can't speak for Taigu or Jundo, but I suspect there are many times when their practice is constantly beset by thought. Mine certainly is! All ideas of 'better' or 'worse' simply make no sense in a practice confined to each moment as it happens.
When you sit, your mind will likely begin taking you on a 'ride' somewhere. At first, this is usually a 'mental' ride - a 'grocery list' of things your discursive mind helpfully tries to bring to your attention. I call it a 'ride' because you are invested in these little trips. You may find them important or tiresome or worrisome - but in some way, your attention is drawn into them to the inevitable exclusion of immediate experience. You are involved in these trips. As soon as you realize that you are involved in a trip - that your identity is invested in a trip, you bring your awareness back to 'right here' (which is really 'nowhere in particular'). You 'get off the ride'. When you begin Shikantaza, you'll find that your helpful mind almost immediately begins to take you on another little 'trip'. Your attention and identity become drawn to another discursive thought pattern. This is actually a good thing in a way, because only by having your mind take these little trips can you realize how your attention and identity are getting sucked into them! The dropping can only happen when there is something to drop, and the activity of 'dropping and coming back' can really only be cultivated in a delusive and confused mind. The important thing is to realize you are being taken for a ride. Truly, simply noticing it typically causes your identity to dissolve. It's very important that you don't make the 'activity of dropping' yet ANOTHER ride.. There is no 'dropper'.
Once you begin to do this 'dropping', something very tricky will begin to happen. Typically, your mind will choose more and more 'rides' that have an emotional quality to them. Fear, greed, lust, outright terror, etc. The egoic mind adds more 'firepower' to the trick, but it is essentially the same trick. Try to keep some compassion for your egoic mind! It really does think it is trying to help! It can be a bit more difficult to 'drop' emotionally charged 'rides'. Sometimes, you may have to take some 'off cushion' time to truly investigate these emotionally charged thought patterns. My favorite thing for this is Byron Katie's 'Work' - because it actively shows the egoic mind the errors in its assumptions - that is, it shows you how much you really don't know - but this is not done in interruption of your practice. If you are sitting and fear comes, notice the 'stickiness' of the emotion/thought, and drop it to the best of your ability. If it does not 'drop', then sit with it with awareness. You may discover some true strength in being able to sit in the face of absolute terror! Doing just this helped me overcome a rapidly developing Panic Disorder.
Lastly, if your dropping reflex becomes very quick - if you become aware of the 'rides' before they even start and drop them immediately, you may experience thoughts nonetheless - floating, sideways, etherial....but they are disidentified and they are part of the now - they do not form an identity and pull you from 'right here, right now' - they become PART of 'right here, right now' (they always were, but your mind screened out the now in favor of a particular thought pattern). At this point, the thoughts actually may become less frequent. Without the energy of identification, there is much less energy for thought creation. Nonetheless, an unidentifiedawareness begins to become known to you. It has always been there. It is not 'developed' by sitting Shikantaza. As a matter of fact, identification is a FUNCTION of the thought trips - there is no identity aside from them.
I hope that helped more than it confused. Shikantaza really is much different from 'mindfulness' practice as I know it.
Jundo or Taigu will be along to correct me shortly (I hope).
Chet