While the story seems to have been forgotten to memory, The New York Times printed in 1988 an anecdote from a Jewish survivor of the Buchenwald concentration camp:
''In the camp there was a cage with a bear and an eagle,'' he said. ''Every day they would throw a Jew in there. The bear would tear him apart and the eagle would pick at his bones.'' ''But that's unbelievable,'' whispered a visitor. ''It is unbelievable,'' said Mr.Hubert, ''but it happened.''
I guess what I'm asking is: "Did it happen?" Any attempt to research this story only brings back this initial primary claim by Mr Hubert, and any attempt to bring this up for further inquiry in Holocaust-related research channels is shut down without further comment (I imagine it's because of how controversial the claim is, given that a bear gaining weight for hibernation can't consume more than 90lbs a day max)
No credible source has ever claimed such a thing, so the answer is simple: no, of course the Nazis did not "really keep a bear and eagle at Buchenwald Concentration Camp for the purpose of eliminating Jewish prisoners". People who were murdered directly in Buchenwald (of all backgrounds) died from shootings, hangings and certain brutal experiments.
It is regrettable that the NYT printed Hubert's claim, based either on rumors or on imagination, without a further comment or fact-checking, but you won't find such a claim in any respected history book and no such claim is known from the other camp survivors.
The claim, although wholly irrelevant to the historicity of the Holocaust, is used for propaganda purposes by Holocaust deniers, who usually make screenshots of the article and highlight Hubert's claim to post it to Twitter, Facebook etc. - that's their weird idea of effective propaganda. They seem to be under an impression that it they discredit some people making claims about the Holocaust, they thereby discredit all Holocaust witnesses. Yes, truly, Holocaust deniers are that dumb. You can read more about this canard here: http://holocaustcontroversies.blogspot.com/2017/05/rebutting-twitter-denial-most-popular.html#witnesses
So this may explain a reaction to your question elsewhere.
Now, the origin of the claim is not hard to guess - the SS kept a zoo for themselves in the camp, with a special bear kennel, here you can see an SS postcard with two bears from that zoo: https://www.sueddeutsche.de/politik/konzentrationslager-das-grauen-des-kz-buchenwald-1.442969-6 And here is an in-depth article in German: https://jungle.world/artikel/2015/16/den-tieren-geht-es-gut As one can see, there were also eagles in the so-called Falkenhof.
So, as I wrote above, if Hubert didn't outright invent the story, he had probably heard a rumor based on the existence of the zoo. This is similar to the "Auschwitz rollercoaster" denial claim I have previously addressed here: https://www.reddit.com/r/AskHistorians/comments/csxu8o/comment/exhpupg?context=3
