Yes, the speech occurred. On 27 July, 1900, Kaiser Wilhelm II addressed a contingent of German marines that were going to be sent to China to help suppress the Boxer Rebellion. In front of these men, as part of his speech he said,

Should you encounter the enemy, he will be defeated! No quarter will be given! Prisoners will not be taken! Whoever falls into your hands is forfeited. Just as a thousand years ago the Huns under their King Attila made a name for themselves, one that even today makes them seem mighty in history and legend, may the name German be affirmed by you in such a way in China that no Chinese will ever again dare to look cross-eyed at a German.

(full translated text can be found here)

The passage I quoted was seen as scandalous by members of the German government. Foreign Secretary Bernhard von Bülow, who would become Chancellor later in the year, attempted to suppress the full text of the speech, as he and others felt that the actual text would cause outrage internationally and domestically. For all of his efforts, primarily by giving edited copies of the speech that omitted the above passage to the press, a journalist was able to obtain a full copy and publish it in the Weser-Zeitung on 29 July.

This speech has, in popular memory, formed the basis for the nickname "Huns" given to the Germans in both World Wars. While this is a popular explanation for the epithet, some research has argued that Hun was a more general term in use in the United Kingdom before the war, and that it was recontextualized with the start of World War One. The main thrust of the argument made by Andreas Musolff in this paper is twofold. Firstly, he argues that the usage of the term hun in the speech is very different from the usage of it by the Allies. Wilhelm was attempting to compare Germany to a warrior-nation, that the soldiers he was addressing should fight bravely like the ancient warriors. The Allies on the other hand were attempting to paint the Germans as a beastly "other".

The most important part of his argument is this:

In nineteenth century English it had become a floating signifier, together with Vandals and Goths, to designate and stigmatize any group suspected of “barbarity”, lack of “civilization” and ignorance of/resistance to Western “cultural” standards.

He argues that "Hun" had been in usage far before Kaiser Wilhelm's speech, and was used in both English and German to denounce an "other".

The biggest reason I can think of (outside of apologia for Imperial Germany) for why someone would believe that the speech did not actually have the quoted portion is a lack of contemporary audio evidence. In 2012 a phonograph recording of the speech supposedly surfaced (Der Speigel has the recording available here). From what I understand, the identity of the speaker in the recording is not firm - nor is the exact date (estimates range from 1900-1906). As well, judging by other confirmed audio recordings of the Kaiser, this does not sound much like him to me. Even without audio recorded while he gave the speech, we have plenty of other sources which do state that the Kaiser gave the speech in its infamous form.