Fascinating question! This is going to be US-focused, as that's what I'm familiar with. If I have time tomorrow I may add some stuff from other angles.

By September 1939, when the war began, certainly the persecution of the Jews in Germany was known. A little less than a year earlier, in November 1938, Kristallnacht had occurred, a massive nationwide pogrom sanctioned by the Nazi government in which at least 91 (but almost certainly far more) Jews had been killed, hundreds more had been injured and/or raped, and more than 250 synagogues and about 2500 Jewish-owned businesses had been destroyed; immediately after, more than 30,000 Jews were arrested and put into concentration camps. But that was only the most violent manifestation- already the Nuremberg Laws and other laws had greatly limited Jewish freedoms in Nazi Germany, and smaller scale persecutions had been rampant. While these issues were not necessarily given tremendous coverage, they were definitely widely known.

By Dec 1941, when the US entered WWII, the full extent of Nazi actions against the Jews was not yet known. Of course, much of it hadn't necessarily happened yet- the Final Solution wasn't decided until a month later, leading to massive deportations of Jews in ghettos to the six major death camps, for example- but already the Einsatzgruppen were committing mass shootings of Jews in areas newly conquered from the USSR, and Jews were living in ghettos in horrendous conditions. Of this relatively little was known- it was often known that Jews were living in ghettos but not necessarily how dire the conditions of these ghettos were. (I once saw a contemporary account of the ghetto in an American Jewish publication which made it seem as though the biggest issues in the ghettos were insanitary conditions leading to deaths among children and the elderly as well as a lack of Jewish education. They obviously were very underinformed as to the true gravity of the situation.)

Even before the full enormity of the situation was known- that there wasn't just murder of Jews, but an actual campaign for the destruction of the Jewish people- Jewish boys were still very motivated by ideology as they fought in the war- what they DID know about what was going on in Europe was bad enough. While Jewish boys didn't necessarily run to enlist, and instead often did wait for the draft, once the draft notice came they did not shirk their duties, and often even boys who had exemptions from the draft (such as divinity students) often volunteered as ambulance drivers or air raid wardens to assist the war effort. This began a full year before information about the Nazi annihilation campaign was publicized in the US- and was first confirmed by the State Department- in November/December 1942. These Jewish soldiers were thus highly motivated by their desire to defeat the Germans- which was often at odds with their fellow soldiers, who considered their status in Europe as merely supporting the British and felt that the real war was happening against Japan. This was only one way in which Jewish soldiers often found themselves at conflict with fellow soldiers; Jewish soldiers later reported endemic casual antisemitism among their comrades in arms, often fueled by ignorance.

By the end of 1942, when the full extent of the Final Solution became widely known, US troops weren't in Europe- they were in North Africa, far from the concentration camps. However, once they did make it to Europe, Jewish troops were numerous among them, from D-Day on forward (my great-uncle's brother was a Jewish soldier who served as part of naval support for D-Day). I don't know of any specific effort to keep them away from the front or from areas where they might be captured by Germans, and indeed some were. Jewish soldiers were marked by a letter H (for "Hebrew") on their dog tags, and there were certainly discussions by Jewish soldiers whether to remove their dog tags in the event that they were captured. Many decided that they would not, but others did- with one soldier throwing away his dog tags but finding that the last name Shapiro told the whole story anyway. In some cases, the rest of a Jewish soldier's comrades in arms might remove their dog tags as well in solidarity. Some soldiers tried to forestall the issue by etching over or erasing the H on their tags.

Then came the question of whether to, when captured, step forward and admit to one's Judaism when the question almost inevitably arose upon interrogation. Functionally, this rarely made a difference, though some Jews who lied later felt a tremendous amount of guilt about having done so. Jewish soldiers were by and large treated in much the same way as their non-Jewish fellow POWs*. However, for about 300 Jewish soldiers captured in 1944 following the Battle of the Bulge (as well as several dozen other "undesirables" among the POWs), the fact that they were Jewish made a big difference. That said, as with the unfortunate Shapiro mentioned above, one didn't need to declare himself to be Jewish in order to be singled out by the Germans; they were quite happy to go through the POWs and isolate those with Jewish-sounding last names. These 350 soldiers were sent to Berga, a subcamp of Buchenwald where they were forced to hollow out a mountain for use as a German factory, and were later led on a death march. 20% of these American POWs were dead by April 1945.

Some of these Jewish soldiers actually were at something of an advantage when they were captured by the Germans- many of them spoke Yiddish, which while not identical to German was similar enough that they were often able to make themselves understood. One Jewish POW who had hidden his Jewish identity used his Yiddish to help translate between his captors and his fellow soldiers. Sometimes, Yiddish-speaking soldiers were even used to translate for German POWs in Allied camps, both in Europe and in the US (my grandfather's friend was a camp guard in a POW camp at Fort Drum, NY, where he spoke to the German POWs in Yiddish). Of course, later on, the ability of these Jewish soldiers to speak Yiddish, and the fact that they were on the spot in Europe, proved tragically useful as the concentration camps were being liberated, and these soldiers were able to communicate with the survivors. It was here that these soldiers, who had been fighting for the sake of their Jewish brethren in Europe, began to fully understand the enormity of what had taken place there and to see the destruction of their people (not just in the abstract sense, but often their actual family members, as some of them were relatively recent immigrants to the US)- but also to participate in the renewal of Judaism in the survivors through leading prayer services with Jewish chaplains, lending Jewish ritual items, providing kosher food, and reminding the survivors that thriving Jewish communities still existed elsewhere and the Jewish people had not been destroyed. While occasionally non-Jewish soldiers would accuse the Jewish soldiers of favoritism in their treatment of the Jewish survivors, this did not stop them.

If you're interested in this topic, I highly recommend GI Jews by Deborah Dash Moore. It's a fascinating look at the wartime experiences of Jewish soldiers in the US Army told through their own eyes, specifically in the European theater.

*This was true for Allied soldiers EXCEPT for Polish-Jewish and Soviet-Jewish soldiers, who were treated with far greater cruelty.