As others have already pointed out, the starting point here must be to differentiate between Death Camps and Concentration Camps. /u/Kugelfang52 provided an excellent run-down in his linked answer, which they also mentioned below.

With Auschwitz (and to a certain extend I'd also count Majdanek) as the only camps that served as both extermination and concentration camps, the public perception of those two has -- through Auschwitz becoming the "example" camp -- been mixed up in popular reception. As my esteemed colleague writes in their linked answer, Concentration Camps served as interment and imprisonment camps. With the latter it is often asserted that they were somehow inspired by either the camps at the trail of tears of the British camps in South Africa. For both of this, there is little evidence, beyond in the latter case, the name.

Concentration Camps in their function and set-up do have a firm tradition in German history of the 20th century. The first such camps appeared under the same name at the Krupp company facilities in Barmen, where left-wing workers were interned during the First World War. This system of establishing make-shift camps for political opponents and people deemed problematic found wide-spread usage in Germany from the First World War onwards. Nicholaus Wachsmann in his history of the Concentration Camps firmly asserts that the traditions of the Nazi concentration camps including the usage of camps, the military organization and demeanor, the prisoners being forced to work has firm roots within the German penal system and the German army. According to him, the concentration camps represent a militarization in the style of WWI German army traditions in the penal system, the prisons set up the reformers of Weimar being seen by the Nazis as too weak and friendly. In essence, the people setting up the CC system, most notably the Inspector of the Concentration Camps Theodor Eicke, referred back to his own experiences in the German Army in WWI and its penal system when setting up the camps.

When it comes to traditions going back further, it becomes difficult. As my colleague /u/Bernardito points out on a regular basis, the concentration camps of the Boer wars in South Africa bear only passing resemblance to the Nazi concentration camps in that they were not part of a policy of systematic cruelty and that the conditions in these camps were not result of a governmental approved policy but rather of local conditions. It is a reasonable assumption though that the name does indeed derive from these camps since the Boer Wars and sympathy for the Boer cause was something that was very closely followed etc. in Germany at the time it occurred. This does not extend beyond the name however, since the usage of camps for such purposes was something the Germans were well acquainted with through their own colonial history.

When we talk about the first genocide of the 20th century being the Armenian genocide in the Ottoman empire, we tend to overlook Imperial Germany's treatment of the Herero and Nama in 1904. The colonial administration of German Southwest Africa not only committed what amounts to genocide against these people but also built several camps that were called concentration camps or were used in a similar way to the usage of camps by colonial administrations all over Africa. The most famous one probably was Shark Island (back then Lüderitz Bay), where 2000 Nama and their families were interned and forced to work for the Germans.

So, when we trace this particular usage of camps, we can point to a German tradition which would have been much more familiar to historical actors like Himmler and Eicke than the much further removed policy by Andrew Jackson. At the same time, we need to be careful not to overinterpret these traditions. While the name of Concentration Camps has been in usage in Germany since the Boer War, the idea of using camps for political opponents is neither a new one nor a particularly revolutionary one by 1933 or even in the Trail of Tears.

The assertion that the Nazis were inspired by others in their camp system is further complicated by the fact that they themselves never really refer to it that way. As has been pointed out Hitler had read Karl May and most of the Nazi regime had a passing familiarity with the treatment of Native Americans in the US but in a political context, this only comes up when the Nazis talk about their own conquests. Both Mark Mazower and Wendy Lower write about the Nazi occupation of Europe in terms of a colonial project. This is backed up by quotes such as Hitler referring to Ukraine as Germany's "California" in the sense of a golden place of settlement and riches unencumbered by anyone worth living actually living there. Another and much more frequent reference in this context however, is British India, which the Nazis imagine as this place of thousands of natives run by a small elite of British overlords without many problems.

The problem in the end is that beyond these passing references influenced by German pop culture rather than historical reality, we have no way to assert that the Nazi leadership was even aware of the Trail of Tears and the usage of camps during it. With a much closer tradition of camp usage of their own and it being unclear whether they were familiar with it at all, I think the idea that Jackson and his policies inspried the Nazis even in the remotest sense is one that does not hold up to close scrutiny.

Sources:

• Nicholaus Wachsmann: KL. A History of the Nazi Concentration Camps.

• Mark Mazower: Hitler's Empire.

• Wendy Lower: Nazi Empire-Building and the Holocaust in the Ukraine.

• Jürgen Zimmerer, Joachim Zeller: Völkermord in Deutsch-Südwestafrika: der Kolonialkrieg (1904 - 1908) in Namibia und seine Folgen.

• Jonas Kreienbaum: „Ein trauriges Fiasko“. Koloniale Konzentrationslager im südlichen Afrika 1900–1908.

Concentration camps had been set up in South Africa during Kitchener's 'Scorched Earth Policy' during the South African Boer War in the late 20th Century. Whether that policy was influenced by Jackson, or in turn influenced Himmler is an interesting idea - but at the very least it highlights that concentration camps were policy in various governments around the world.

One other thing to consider is that the origin plan of the Nazi party wasn't death camps - at least according to Timothy Snyder's Causes of the Holocaust - but rather the deportation of all Jews from the new Germany. With the events of SS St Louis and England entering the war, along with a longer than expected campaign in Russia, the Nazis were running out of options to dispose of their 'Jewish problem' so they installed Gas Chambers at the camps they already had set up and proceeded to go about murdering them. It was supposedly the easier option.

This is what Snyder wrote:

The crucial change in policy, from deportation to killing, took place in Eastern Europe, where the German racial empire was to arise, where most Jews lived, and where the technologies of mass killing were institutionalised. [W]e know that until 1941 German policy was to deport Jews, and that until 1939 this policy was implemented with some measure of success. German policy from beginning to end was to eliminate all Jews under German control; the crucial question … is when and how this became a policy to kill them.

Not sure if this is helpful, but I don't know much about Jackson or his policies - but I do know a little bit about Himmler and a little more about Kitchener. Let me know if you want more sources or info.

Quite interested in this as it's one of those things I've often heard repeated (along with the Boer War comparison, which seems more likely as it attracted significant public outcry in German) - and may have indeed repeated it myself. Having had a quick look now, I can only find passing references and I'm struggling to pin down the source.

In the meme I recall, it was Hitler rather than Himmler who 'got the idea' from the US (which is problematic because it assigns one person 'creative ownership' of the concentration camp system), an interesting side-point is that Native Americans played a minor, but significant, role in German propaganda.

Karl May novels, peddling a romanticised noble savage myth in pulp adventure format, were hugely popular in German well into the Nazi era. Hitler was a fan - in 1944 he had 300,000 of May's books printed and distributed to the troops, and kept vellum-bound copies in his library.

Klaus P. Fischer posits in Hitler and America that a lot of Hitler's key themes are echoed Karl May, with Lebensraum being Manifest Destiny run through the filter of National Socialism, and square-jawed supermen overcoming the "savages" fitting neatly with the Nazi self-image.

I'd be curious to learn if the idea that the Nazis were directly inspired by the the Trail of Tears was a post-war myth built on the Nazi fetish for the Old West, as much as I'd be interested to the find out the extent to which it informed the concentration camp system.

Apologies for the segue. If it's too off-topic, please remove.