When folks are told that Zen involves non-conceptual thinking they often have problems wrapping there mind around it. They can't "conceive " of it. Yes that was a pun.
Part of the problem is many people think of a concept as the basic unit of thought. The smallest unit of cognition. In Buddhism we call the smallest unit of cognition a mind, not a concept.
Buddha taught that we are all deluded, we live in a world of delusion. He didn't say the world wasn't real, just that we were ignorant of its true nature and how it and we really existed.
The modern philosopher Grant Evans put it this way:
" Behind the idea of a system of beliefs lies that of a system of concepts, whose structure determines the inferential properties of the beliefs."
Our training, world experience and socialization shapes the world we live in, our Samsara.
On a more particle scale if we are dealing with the concept of a chair I think things may become easier to understand. We encounter an object, we perceive it and we automatically run through our concepts to see into which it fits. There are many types , styles designs and forms of a "chair", but usually we can agree on whether something is or isn't qualified to be called a "chair". We look to see if it fits into or out of our concept of a chair.
We pull away from the very idea of non-conceptual thinking in fear. Our world is based upon our concepts you say, without them all would be lost. I tell you laugh in the face of this silliness.
You have non-conceptual thought every time you encounter something new, every time you can't fit a new object or idea into the list of concepts you have generated and stored up during your life time.
Now is when I bring out D.T. Suzuki's "Zen Mind, beginners mind" Just what dose that mean?
OK, you walk into a room, there is something on the table like nothing you have ever encountered before. the best you can do is decide its a material object. You reach out and your hand passes right through it. Now you have no concept for it at all. Your mind churns to find one for it. I walk in behind you and say "I see you have found my holographic atom sorter."
I have started to pass along to you the concept of the holographic atom sorter, and boom your back in conceptual thought land again.
In truth there is non-conceptual content to every thought you have. Zen can show that to you it can train you to be aware of it to experience it. To bring you back to the beginners mind.
In the end we must leave all the conceptual "belief systems" behind, even the Buddhist ones. We cross the river and leave the raft that is the Dharma on that far shore.