Humphreys spent a lot of time trying to cast doubt on the reliability of Christian sources. He kept referring to Christian documents, like the gospels, as biased or unreliable in some other manner. And he kept changing his standards during the discussion, which sometimes makes it difficult to evaluate his position. At one point, he asks for sources other than the gospels. At another point, he asks for non-Biblical sources. Then he asks for secular sources. To make matters worse, he would sometimes ask for such sources after McDowell had already provided some. When Josephus, who meets Humphreys' definition of a secular source, was brought up again after initially being ignored by Humphreys, he objected that the passage in Josephus is so short, was preserved by Christian sources, etc. At one point, around the 47:30 mark in the program, Humphreys refers to how having "a secular source to back it up" would make McDowell's case far more convincing. He refers to "how very convincing that might be" if we had a secular historian somewhere in the Roman empire who commented on the death of an apostle. He goes on to refer to how we have to rely on Christian sources instead, apparently implying that there aren't any secular sources to support a traditional Christian view of the death of any of the apostles. Though he added the "historian" qualifier at one point, as if the secular source in question has to be a historian, most of the time he doesn't include or imply such a qualifier. So, it seems that he's arguing that we only have Christian sources to go by.
At his web site, he goes as far as to say:
There is NO corroborating evidence for the existence of the twelve Apostles and absolutely NO evidence for the colourful variety of martyrs' deaths they supposedly experienced. The Bible itself actually mentions the death of only two apostles, a James who was put to death by Herod Agrippa (see James for a discussion of this tricky character) and the nasty Judas Iscariot (see below), who gets several deaths because he's the bad guy.
Humphreys is wrong. A few years ago, I wrote an article that addresses what ancient non-Christian sources tell us about the death of the apostles. One of the points I make there is that the relevant sources go beyond Humphreys' secular category. Some of the ancient heretics had an interest in denying the martyrdom of one or more of the apostles. Think, for example, of how many ancient heretical groups were opposed to one or more of the apostles and therefore had a motive to deny accounts of those apostles' deaths that made them look good. It wouldn't make sense to exclude such heretical sources just because they aren't secular by Humphreys' standard. It seems that the principle Humphreys is getting at is that we should be looking for sources who don't have a bias toward affirming a traditional Christian view of an apostle's death. But secular sources aren't the only ones who can fall into that category.
In his book on the martyrdom of the apostles, McDowell gives some examples of ancient Christian sources saying or implying that various apostles didn't die as martyrs. And many modern Christians do the same. It's common for Christians to say that the apostle John died of natural causes, for example (though I disagree). So, having a Christian bias doesn't require that you believe that a given apostle died as a martyr. And it's somewhat common, not just a rare occurrence, for a Christian to say that one or more of the apostles didn't die as a martyr or that the evidence for an apostolic martyrdom is weak or too ambiguous to justify a conclusion.
Anybody who's interested in doing more research on the death of the apostles can read my series on the topic here. You could also read McDowell's book, The Fate Of The Apostles (Burlington, Vermont: Ashgate, 2015).