You open yourself to the argument that if the long day can be shown not to have happened, God, Religion, Church are refuted, no?
And the short answer to Charlie Totten is: if his explanation were correct in modern scientific terms (although it wasn't correct even according to the science of his say) then it would be orthodoxy and we'd know when and how the earth ceased rotating for 23 hours and a bit. But the physical consequences would have been be utterly catastrophic, utterly unmissable. Picture for example the momentum of the water in the oceans, traveling eastward with the earth's rotation at 1037 mph at the equator, continuing east while the earth's rotation suddenly ceases, and humans, animals and objects all flung eastward at such speeds as the ground under them simply stopped. Even Joshua would have blinked at such a massacre.
I found this summary of Totten's book by
Robert C. Newman, a christian and a scientist:
Reading Totten's book brought another surprise: [Totten's earlier] dramatic story of a skeptic convinced does not appear. Instead, Totten himself, a non-skeptic all along, seeks to show that a total of 24 hours is missing from past time, of which 23 hours and 20 minutes were lost in Joshua's day and 40 minutes at the time of Hezekiah. Totten does not actually reproduce the calculations by which he seeks to prove his 'case'; he merely gives the results. On pages 39, 59, and 61 of the third edition, the fact emerges that Totten relies on an assumed date of creation – the autumnal equinox, September 22, 4000 BC (p.61) – as the 'known' fixed point before the long day of Joshua. Taking the first day of creation to be a Sunday, by his understanding of Scripture, and finding that by calculating back from the present, September 22, 4000 BC would fall on a Monday, Totten concludes, "It can come so by no possible mathematics without the interpolation or 'intercalation' of exactly 24 hours" (p.59).
Not much sign of science in Totten then, you'd have to agree.