At first sight this claim is not unreasonable. Theories seem to me always based on some beliefs, albeit reasonable beliefs. For example belief in the validity of the logic used. But why should these beliefs be religious? Can Clouser proof the claim of the hidden role of religious belief?
Clouser was a professor of philosophy and religion. His book is therefore systematic, quite profound, focusing on fundamentals and therefore not for the faint of heart. Although he is clearly a Christian, he is not a 'fundamentalist' which is according to him someone who holds the 'encyclopedial assumption'. This assumption entails 'that sacred Scripture contains inspired and thus infallible statements about virtually every conceivable subject matter'. This was centuries ago a common view point. In stead, Clouser believes in a more subtle but very fundamental influence: Scripture does not dictate the contents of theories directly, but provides the basic assumption(s) they are based on. Hence the 'hidden role' of religious belief. Throughout the book it becomes apparent that this belief should better come from the Bible, according to Clouser.
How does Clouser proceed to prove the claim of the (sub) title? He starts out with some pleasantly clear, concise definitions. The one of 'religious belief' is a case in point: 'a religious believe is any belief in something or other as divine.' In this 'divine' means 'having the status of not depending on anything else', in other words, being self-existent. The Christian God is an (the only?) example. Three chapters are devoted to showing that theories in mathematics, physics and psychology are all based on a religious belief. That is, pagan beliefs, according to – again - his definition: a pagan belief is a belief that the divine is some part of the (God-)created universe, in other word, 'reality'. He gives as an example in physics Einstein's theoretical foundation. That is his (supposed) believe that the laws of logic and mathematics govern all reality (including human thought). This means that they are self-existent and thus divine. Hence Einstein's view is religious. In addition, because logic and mathematics are part of reality his religious belief is pagan.
In fact, this ends the part of the book 'proving' the claim of the (sub) title. Is the proof valid? I do not think so. A major weak point is that the claim is based on self devised definitions. One can proof anything from properly constructed definitions. Take the definition of the divine being self-existent: it is questionable whether there are non-hypothetical independent entities at all (of course the Christian God is an example according to Clouser). The 'religious' creeps in through this questionable definition.
After having shown that current theories make the wrong – pagan - assumptions Clouser devotes later chapters to biblical theories of reality and society. He skips mathematical, physical and psychological theories for two reasons: he does not have enough knowledge and his own 'theory of reality' comes into its own in social theories. The skipping further weakens his position.
In the afterword Clouser questions whether his religious, Biblical view does not divide people. He thinks this is not the case, again for two reasons. The first is that giving fundamental reasons for differences between theories will not lead to intolerance. Secondly, these reasons will bring forth a fruitful communication. What he does not mention is that he says that the Christian Bible provides the best ground for theories to be based on, which of course has the potential to divide people.
Clouser states that 'Having the right God is basic to all truth'. This looks to me as a basic assumption of himself. It seems to be the inspiration of his theory and the book in which it is described. A book which is unusual, faulty and nevertheless (or therefore?) thought provoking.