What is considered to be free choice in the context of "choosing to go to hell"? Let me preface this by stating that I received a Christian education and have seen the idea that individuals choose to go to hell (rather than God sending them there), mostly from a vaguely evangelical context. This statement bothers me for various reasons, I won't go into, but it strikes me as rather vague for the following reason.

Suppose person X hears about Christianity, but for some reason, does not accept it. There are various reasons for why this might happen (it's not convincing enough, for instance), but one that strikes me as particularly "problematic" (and not just so from my perspective as an atheist) is one where X hears about Christianity from some very poor representatives of the religion (e.g. Christians who have decimated the people of X's religion of ethnicity). As a result, because of X's poor experiences, he refuses to convert even when he is told about Christianity. (There's some reference to this sort of scenario in some accounts of the conquest of Latin America.)

Does this count as a scenario where X chooses to go to hell, as per the formulations of people who believe in this account of where people end up in the afterlife? More broadly, what is meant by "choice" in this context of choosing to believe in God?
The statement of "choosing to go to hell" is an over-simplification that really is an example of intellectual laziness, and is a great example of the difference between the "true theology" of a group and the type of pop-theology that springs up and causes great confusion.

Looking at it from any side, saying that people choose to go to hell does not make sense.

First, there are no passages of scripture that specifically mentions a person choosing to go to hell. Rather, multiple passages mention an action by God causing the separation of people, from the allegories of separating goats from sheep, or wheat from chaff, to the judgement scene in Revelations 20. Note that the separation is more of an acknowledgement of a preexisting distinction, but still only happens at the action of God. Then there is Matthew 7:21, in which it appears there is a person who wants entry to heaven, but God turns them away because they did not approach the correct way. Obviously not a choosing of hell, but rather a failure to choose heaven the right way.

Second, even just a casual examination of those who are not "born again", or those who have not "accepted Jesus", clearly shows that they have not seen a black and white choice between a choosing of heaven and a choosing of hell. Almost nobody ever believes in the existence of hell and then deliberately chooses it. As you said in your example, normally people just don't believe; not a choosing for one side versus the other, but rather an absence of choosing either.

So yes, according to the normal Evangelical interpretation there is a choice, but it is a choice to believe in and follow Jesus versus a choice not to believe in Him or not to follow Him. And yes, a choice to follow Jesus results in being allowed into Heaven, whereas a choice to not follow Him results in not being allowed into Heaven, and instead being sent to Hell, but that is a consequence, not the actual choice.

Unfortunately some people boil this down into the statement that a person is choosing to go to hell when they are actually choosing to not believe in Jesus (or at least the Evangelical interpretations).

All denominations/movements, and religions for that matter, are filled with pop-theology - the latest thoughts that spring up, which sound good to those sharing it, but don't hold up under scrutiny based on the group's base theology.