Well, it depends on how you define "non-Jews." Many people who had some Christian ancestors and some Jewish ancestors, and even people who were married to Jews (or people with Jewish ancestors), were often sent summons to register as Jews and take part in "resettlement". The same goes for people with Jewish parents, grandparents, or great grandparents who did not actively practice Judaism. Once a person was in a concentration camp, it was nearly impossible for them to appeal the decision. However, there are a few documented cases of Christian relatives of people labelled "Jews" (for example, if one person with Christian parents had a cousin who had one Christian parent and one Jewish parent) appealing to the government to protect their relatives, or even getting a person removed from a camp. However, this was relatively rare.

The core issue here is how one defines a "Jew." Someone who practices Judaism? Someone who has a certain number of Jewish parents or grandparents? Someone who looks stereotypically Jewish? Even someone who has "too many" Jewish friends or relatives? These were issues that were hotly contested even during the Third Reich.

Other things that are important to remember: it wasn't just Jews who were sent to concentration camps. Roma (commonly referred to as Gypsies), the mentally and physically handicapped, LGBTQ individuals, petty criminals, and political dissidents were all murdered or sent to concentration camps by the Nazis.

Another VERY important thing to remember is that concentration camps were not the fate of most of the Jews murdered by the Nazis. Mobile death squads often went from town to town (mostly in Eastern Europe) and murdered those labelled to be Jews by shooting them and burying them in mass graves (sometimes making them dig their own graves), or locking them in a synagogue and setting it on fire. There's even a case of a town in Poland where half of the population (Christians) murdered the Jewish half of the population without any apparent pressure or incentive from the government.

There are a lot of things at play when we consider 'the Holocaust': first of all that at the time, antisemitism was ingrained in many parts of the world, not just Germany: the UK and US included! A lot of people at the time thought Hitler had the right idea about the Jews, and just didn't like his aggressive expansion policies. Stopping the murder of millions of innocents was NOT the primary motivating factor for the Allied powers during the war. The Holocaust was considered one small event in the greater context of a war. Secondly, it wasn't just Jews being murdered, and they were certainly not all murdered in concentration camps. Finally, that the definition a "Jew" was something that was constantly in flux. There were well-known Jews who were spared for a myriad of economic, social, and political reasons, often because they weren't considered to be like "most other Jews," and sometimes because they were put in positions of power over more suffering Jews in order to aid the transition from normalcy to increasing oppression to ghetto to the camps or murder. Was being a Jew a matter of race, a matter of religion, a matter of appearance or mannerisms, a matter of economics? Even the fiercest Nazi ideologues didn't have a consistent definition.

Sources:

Saul Friedlander, Nazi Germany and the Jews, Volume 1

Rory Yeomans and Anton Weiss-Wendt, Racial Science in Hitler’s New Europe, 1938-1945

Alan E. Steinweis, Studying the Jew: Scholarly Antisemitism in Nazi Germany

Marion Kaplan, Between Dignity and Despair

Jan Gross, Neighbors: The Destruction of the Jewish Community in Jedwabne, Poland