True Christians must think independently.In fact, since the time of Christ that independent thinking separated them from those who deviated from the teachings of Christ which caused and still causes intense persecution. Jesus was certainly an independent thinker separating himself from the Jewish traditions and wrong doings of the then clergy and clearly aligning himself with the instructions of his Heavenly Father. He was persecuted. His followers who held to what they were taught were also severely.persecuted.
I had a fabulous history teacher in high school who insisted on independent thinking. Her exams consisted of 10 multiple choice questions or fill in the blanks worth 2 points each. The remainder of the exam were essay questions. Only four were given - we had to choose two. Within each of our choices we had to explain the political, social and economic impact of the event and how it effected the outcome of a future event. This meant there was no room for memorizing answers - THINKING a thing through to its end was a NECESSITY.
Its true, most people do not think independently Christian or not. Humans have a herd mentality because of the system in which we live.
However, there are a group of Christians who not only think outside the box but deliberately follow in Christ footsteps and follow the pattern of the first century Christians. One of the key admonitions to those who want to be a follower of Christ is that as they study the Bible they are told "DO YOUR RESEARCH." Libraries are filled with secular historical information that verify and further explain Bible accounts and events. (eg. The fall of ancient Babylon through Cyrus of the Medes and Persians) This type of study and research encourages faith in God. Yet, here is the ringer: Most people won't do this because it would interfere with their THINKING about who God is and who Christ Jesus is.
So it is not a matter of so-called Christians not being allowed to think independently - they will not think independently because it would separate them from friends, family, political and in some instances social and economic benefits.