Pure thoughts cannot do that. Physiologically, thoughts are electric and chemical activity in neurons - but even the chemical activity is the same independently of the exact content of the thought. So, if your humans are imagining a geometric figure, that will produce a certain firing pattern of neurons in certain brain areas, such as parts of the visual system - but you cannot make triangles a dangerous figure and leave squares a nondangerous one, since these will trigger roughly the same kind of activity, just in different neurons.
There is a way you could go, but it is through emotions. In the brain, emotions are a kind of neuronal activity, just like the thoughts, but they can also lead to the production of hormones. For example, anxiety is connected to ACTH production in the pituitary gland (which then cascades into the production of other hormones in the body). Frequent anxiety is famously a cause for epigenetic changes, there was a seminal study on rat pups inheriting anxiety from stepmothers, not through genes but through an epigenetic mechanism triggered by absence of calming behavior in their anxious stepmothers.
It is also possible, and common, to trigger emotions through thoughts. If you go to bed thinking that tomorrow, you will have an important exam you haven't learned for, you'll be anxious even though there will be nothing in your physical environment that's different from every other night. So, on that front, you might have some success.
The problem comes, again, from thought type vs. thought content. The brain's activity is localized in different areas when you are thinking about geometric shapes as opposed to listening to a song - but if you were go off that only, you would have to make all thinking about shapes dangerous (or all listening to music). Alternatively, you could say that the thoughts you need cause a certain emotion, and connect the disease to that emotion - but then, it will be triggered by all occasions of having that emotion, not just the ones caused by thoughts about the godly technology.
Since this is fiction and you have genetically engineered humans, you could make it work with some handwavium. You would need to add to the human body a special new emotion, that normal humans don't have. This emotion triggers a new hormone, engineered to have some effect on the human body. And this emotion is so specific that it can only be triggered by having a thought about the godly technology.
A plausible scenario of that kind would be to add a new level of insight. In real humans, the feeling of "X is true" is an emotion - it is just a pattern of brain activity that happens in our emotional centers when we encounter a thought we believe to be true. (Believe as in "faith" and "certainty", not as in supposing it should be true). So, add a new emotion derived from this one, which only happens when the engineered humans recognize not an ordinary truth, but a truth about the divine mechanism of the universe. You can add some graphic description of mythical extase there, that's highly plausible for an emotion. It will trigger your neurotransmitter, which will do its job wherever it needs to - you can actually decide on the duration of the human's life after the insight by picking your preferred cancer.
As a side effect, since our normal emotion of certainty occurs when we think something is true (regardless of whether it's objectively true), this will probably be also the case for the engineered emotion. So for the divine insight, it will also be plausible to not matter if the engineered human had a correct or incorrect insight about the way the gods function - if they live on Olymp, but he wakes up and is struck by the thought "I'm sure they live on Atlantis", the killswitch will also be triggered. That gives you a good deterrent of even trying to have thoughts in this direction, and even if a brave scholar puts up a theory and dies for it several months or years later, his disciples won't know if it was really true.