Did Germans refuse to take Cossacks prisoners in WWI? If so, why?

Cossacks were taken prisoners after that incident. They were not all shot on sight. This single, unsourced line on Wikipedia is not credible, and certainly at least an unsustained vast exaggeration if understood as 'valid description for the entire rest of the war'.

During the war, that is after this incident from 1914, Germany and Austria Hungary took a few prisoners of war, here are some in pictures and lbi.org

^{(imgur is down atm?)}

And these were not the only ones. The central powers had taken enough of them to address the separately in their POW propaganda campaign:

In January 1915, the Austro-Hungarian Ministry for Foreign Affairs was on hand of an exposé including the anticipated advantages that might result from a nationalistic propaganda offensive among Russian POWs. They should have been influenced in a way that matched with the aims of the Danube Monarchy. The definition of goals remained vague. Nevertheless, the propaganda was intended to evoke and strengthen the “national consciousness” of the POWs and to appeal to their “autonomous cultures.” The end goal was to incite independence movements among the “nations” to which the POWs belonged to break away from Russia. The propaganda campaign planned to address Ukrainian and Polish soldiers as well as those from the Baltic regions, Cossacks, Georgians, Tartars and Kalmyks.
_{— Moritz, Verena, Walleczek-Fritz, Julia: Prisoners of War (Austria-Hungary), in: 1914-1918-online. International Encyclopedia of the First World War, ed. by Ute Daniel, Peter Gatrell, Oliver Janz, Heather Jones, Jennifer Keene, Alan Kramer, and Bill Nasson, issued by Freie Universität Berlin, Berlin 2014-10-08. DOI: 10.15463/ie1418.10374.}

As an unsourced entry on a Wikipedia page, chances are high that this is a made up claim. Kryuchkov was from the beginning a figure much exploited for propaganda purposes. In Russia. The fact that 'after that, Cossacks had to change their uniforms' for example is most certainly not because of 'enemy wouldn't take those as prisoners but shoot them'. A dubious claim in itself for sheer plausibility. And ignoring the fact that Cossack cavalry had their heyday in the earliest days of the war, but later were largely dismounted and deployed as infantry…

And while the early raids of such Cossack units seem to have been like the Russian offensive as a whole moderately successful, neither the Austro-Hungarians were much impressed by the such horse tactics, nor later both central powers of the fighting strength of such hors*m*n when suddenly forced to serve as foot soldiers.

Yet in the First World War the Cossack was not particularly highly regarded as a cavalryman by his foes, although he was certainly a match for the Austro-Hungarian. German cavalry was better trained, much better disciplined and incomparably better mounted. And although the strength of the Imperial Russian cavalry arm far exceeded that of the Central Powers and, on paper at least, was truly formidable, it was unsuited to the demands of modern war.
_{— Albert Eaton: "The Cossacks", Men-at-Arms, Osprey Publishing: Reading, 1972.}

If there were some 'irregularities', not 'only' 'harsher treatment' of prisoners, but outright refusal to take prisoners, then it might be similar in scope and reasons as on the Western front with Belgian 'franctireurs': fear and paranoia on behalf of the German soldiers leading to a temporary phenomenon.

Afraid of Russian cruelties, more than 800,000 Germans fled their homes and moved westward.13 Long lines of refugees filled the roads, with carts full of hastily collected luggage and household goods, sometimes even followed by livestock. Occasionally this human traffic hindered the operations of the German defenders. Cossacks sacked and destroyed 34,000 houses. Civilians as well as the general staff wondered if the Russians could be stopped before they overran the whole of Eastern Prussia and perhaps even Silesia. Prittwitz, in a moment of panic, wanted to retreat to the Vistula.[…]

Myths demonising German behaviour – such as the Belgian babies whose hands had been sawn off by German bayonets – flourished in the press and popular imagery. But they often originated with terrified civilian refugees, and government censorship sought to restrain rather than encourage them. Under pressure from disapproval in neutral states, the German government tried to counteract the negative propaganda by conducting its own enquiry. But faced with growing doubts about civilian resistance in 1914, it doctored its official report so as to sustain the original charge. The bitterly contested truth of the German atrocities shows how the laws and norms of war were used both as a measure of real actions and also as a means of condemning the enemy in a conflict that abolished moral neutrality.

Charges of guerrilla resistance marked other invasions, too. As the Russians entered East Prussia in 1914, German refugees related tales of brutal Cossacks and collective reprisals. In fact, two of the worst cases for which clear documentation exists consisted of the reverse – German military depredations against Polish civilians in the towns of Kalisz and Czestochowa, just over the border in Russian Poland. While brutality by Russian troops in East Prussia did occur and was sometimes prompted by accusations of civilian resistance, it was spasmodic and not driven by a Russian delusion of a German ‘people’s war’. Even the Prussian Interior Ministry concluded that panicky German civilians had exaggerated the brutality.
_{— Jay Winter: "Cambridge History fo the First World War. Volume I Global War", Cambridge University Press: Cambridge, New York, 2014.}

And similarly in Austria-Hungary:

Public morale was maintained in part by a steady stream of atrocity stories – later published in two Red Books – concerning Serbian ritual murders of Austrian women and children and Russian Cossack cruelties perpetrated against Habsburg soldiers.

But it is paramount to distinguish between atrocities that did happen and and those that were either made up or embellished and exploited.

In Germany, shortly after the storming of Liège in mid-August 1914, Moltke issued an order establishing capital punishment for persons participating in ‘any form of unjustified war activity’. Such offenders were to be treated as ‘terrorists’. In March 1915 OberOst informed Vienna that it planned to burn two villages in Russian Poland for each German settlement destroyed in East Prussia – much to the delight of the Austrians, who hoped that the Poles would thereby be driven into their camp.

The severity of OberOst’s measures was a direct result of revelations of Russian terror in East Prussia during August and September 1914. Despite Stavka orders to eschew plunder and malicious destruction, Russian rear- guard and train personnel as well as Cossacks undertook a systematic campaign of terror against the German population. Entire villages – Domnau, Abschwangen, Ortelsburg and Bartenstein, among many others – had been burned to the ground after the Battle of Gumbinnen on 20 August.

The Russian retreat after the Battle of Tannenberg exacerbated the civilian horror. The Russians took with them about 10,000 draft-eligible men (but also women) as ‘hostages’; countless bridges as well as rail and communications facilities were destroyed; and factories and utilities installations were rendered inoperable. East Prussian authorities estimated that the Russian occupation armies killed 1620 civilians, destroyed 17,000 buildings, and stole or slaughtered 135,000 horses, 250,000 cows and 200,000 pigs.8 ‘Pillage, like death’, as the historian John Lynn noted of an earlier struggle, ‘arrives hand in hand with war’. The same story held true for Galicia. The Ministry of the Interior at Vienna reported widespread devastation and destruction by the initial wave of ‘undisciplined, robber-like Cossacks’ who had ‘plundered, robbed, killed, and committed innumerable acts of terror’ against the indigenous population, of whom 500,000 had fled westward. More than 2703 square miles of arable land from Brody to Cracow had been ravaged and lay fallow. Seven million farmers were ruined financially and millions of agrarian labourers reduced to beggars. Hundreds of thousands of cattle had been slaughtered by the Russians. Shell craters and abandoned trenches scarred the landscape. Thaddäus von Cienski was but one of countless nobles victimized by the Russians: his ancient castle at Pieniaki had been razed, more than 2471 acres of forest burned and countless villages levelled. ‘The situation of this land is truly hopeless.’ Pogroms had devastated Galicia’s Jewish population of 650,000.
_{— Holger H. HerWig: "The First World War. Germany and Austria- Hungary 1914–1918", Bloomsbury: London, New Delhu, ²2014.}

In light of such events, a single super human hero from the propaganda campaign prompting 'even in warfare illegal behaviour in his opponents' seems like quite a stretch.

I checked the book

"THE COSSACKS AND HIGHLANDERS DURING THE FIRST WORLD WAR," Proceedings of the All-Russian Scientific Conference, Rostov-on-Don, 18–19 September 2014.

The number of cossacks taken prisoners of war during the 1st world war is estimated 6% (pages 33, 125). This was much lower than the number of prisoners of war in other parts of the Russian Imperial Army (for comparison, the number was 35% in the Russian Infantry), but still substantial enough to discredit the Wikipedia page you are referring to, which, most likely, just copied info from some propaganda materials dating to the time of WWI (ditto Sholokhov's novel mentioned in a comment). None of the articles mentions anything along the lines of "Germans and Austrians stopped taking Cossacks prisoners of war." However, it is mentioned that Kurdish troops (serving in the army of the Ottoman empire) did not take prisoners.

One more thing, taken from "Memoirs of the Russian Imperial Army" by general Krasnov (p. 530):

According to Krasnov, German and Austrian military were under the impression that cossacks do not take prisoners and, accordingly, cossacks POWs were subject to harsher treatment. On the next page (531), Krasnov tells a story of a cossack who run from German POW camps three times, was captured all three times; subsequently was interned to Denmark and, eventually, escaped from there as well.