I don't understand... how can evolution 'stop' working on a specific species?
As long as there are people who reproduce successfully and people who don't, then natural selection is at work, or am I missing something?
Even if this guy is full of it, is there any way natural selection would stop working on a species other than that species going extinct?
Not really. Certain species that form spores or other 'stasis' bodies could be said to be outside the reach of evolution if all living examples had died, since the species could not be rightly said to be extinct if the extant spores had the capacity to at some point grow again.
In a similar sense, plants and animals whose seed has gone into seed-banks, where are they are kept in cryogenic or other stasis inducing conditions, if the species subsequently lost all living individuals, could be said to be beyond the reach of natural selection during the gap between the storing of the seed, and the eventual reintroduction of the species using the stored seed.
As a more extreme example, if instead of Scotty getting caught in the transportation buffer in Star Trek, it had been an otherwise extinct species, we could again say it was temporarily not evolving.
All of these examples are effectively identical, and boil down to the species having its reproductive activities placed on hold for a time.
Generally speaking, if a species is reproducing, then it is experiencing evolution. Even a species that reproduces via cloning is subject to the occasional random mutation.
What is probably meant by the people in the original post's references is simply a reduction in the rate of observable non-environmentally induced phenotypic change - since of course, the phenotype is what is generally used to compare fossils, particularly once they pass the threshold of DNA recoverability.
That is, in much the same manner as could be said of coelacanths and other so-called 'living fossils', they claim that humanity has entered a period of low outward change.
Of course, this is manifestly untrue when you consider the relative stature of humans over time, so they must presumably be writing this off to an environmentally induced change, i.e. we are getting more nutrition and health care, and are instead looking for qualitative differences in brain size vs. body mass, or bone structure, etc.
However, even when no visible phenotypic changes are occurring, genotypic changes continue apace - the so-called molecular clock of non-active changes in the base coding of proteins. While useful in biology and constructing biological lines of descent, there is a definite tendency for scientists, particularly geneticists, to refuse this type of change the label of 'evolution,' because the changes so produced are not visible to natural selection (which is why they can continue at a measured and relatively even pace--changes in coding for a given protein that produce an actual change in the protein will either be selected for, or against, but because multiple codons, or base-pair triplets, code for a single amino acid, these can freely change as long as they don't change the amino acid so coded, and the change is invisible to selection because the proteins produced are identical).
In a rather different sense, there are certain species that after existing as sexual species for a period, down-shift into a 'lower evolutionary gear' as it were, by shifting to parthenogenesis, where the species becomes solely female and reproduces by cloning. This drops the rate of production of change on which natural selection has the opportunity to act from the high rate produced by meiosis and recombination down to the much more infrequent rate of random (stochastic) mutation (in the gametic line only, so this is actually even slower than the rate of stochastic mutation in bacteria).
Even though a case could be made that these creatures are evolving more 'slowly' than others, they are in fact still evolving. The bdelloid rotifers diversified into 360 species since they went parthenogenic . . . but like bacteria, they are known to incorporate foreign DNA, so how much of that diversification is due to stochastic mutations and how much to gene-copying is an open question.
Howard, the Grum
urselves, bacteria, and virus. 
