I'm going to try to answer your question from a prescriptive angle. Mods, please let me know if this answer isn't up to community guidelines.
I personally can't think of any examples of Jewish knights or a source that talks about Jewish soldiers, but that doesn't necessarily mean it never happened. To focus on the second half of you question--were there Jewish knights, the prescriptive literature tells us that was impossible for non-Christians to be knights.
The Ordene de Chevalerie, a 13th-century anonymous Old French poem lays out the steps required to make someone a knight. There has been some debate over the reception of this text, but currently, scholars follow Keith Busby's (see below for citations) assessment that it was influential both in France and beyond. It's basically the earliest extant text on the topic and goes through both the ritual to create a knight and what was expected of a man after his ordination. This text, furthermore, explicitly positions knighthood as a Christian institution--indeed, the text explores the tension of what would happen if a non-Christian desired to be a knight.
This poem tells the (fictional) story of a knight named Hugh (transliterated) who was captured by the Muslim sultan, Saladin, while fighting in the Holy Land. Saladin is interested in chivalry and asks Hugh to make him a knight. Hugh replies that it's impossible because not only does he not dress like a Christian warrior but "Car vous estes viex en vo loy / Si n'avés baptesme ne foy" ("because you follow the old law / you haven't been baptized in the faith"). Eventually Hugh is convinced to go through with the ritual, although he refuses to do the accolade--i.e. the part of the ceremony where the newly knighted individual is struck across the face--probably because hitting Saladin in the face seemed like a bad idea, thought also, I think it's an move on the author's part to make sure that the ritual is incomplete, and therefore not legitimate. Though they go through with the ritual, Saladin is not actually a knight according to the text, which highlights the fundamentally Christian nature of knightly duties.
So what does this tell us about the question of Jewish knights? Obviously Saladin is a Muslim, not Jewish. Yet from a medieval Christian perspective, both religions erred in similar ways. While medieval Christians recognized Judaism as a legitimate (if, in their opinion, wrong) religion, they often disregarded Islam as either a heresy or a type of paganism. A lot of medieval texts, including the Ordene, refer to Muslims as "pagans" and about one hundred years later, Dante Alighieri refers to Mohammad as a schismatic (a type of heretic) in Inferno. However medieval Christians recognized that both Muslims and Jews held the Old Testament as a sacred text. The problem with Jews, according to medieval Christians, was that they never moved beyond the Old Testament, while the problem with Muslims was that they did but they got it "wrong."
According to the Ordene, Saladin is excluded from becoming a true knight because he is Muslim--i.e. because he only follows the Old Testament correctly. Following this logic, a Jewish man would also be excluded from knighthood.
It would have, furthermore, been scandalous for a Jew to attain the type of social status held by knights, partially because of threat of social intermingling and the potential for apostasy or "judaizing". The Gesta Philippi Augusti (the Deeds of Philip Augustus), written by the chronicler Rigord at the end of the twelfth-century, shows some of these attitudes. Before expelling the Jews from his lands, King Philip II Augustus of France (the eponymous Philip) was shocked to see the wealth the Jewish community of Paris had achieved: "[the Jewish community of Parish] grew so rich that they claimed as their own almost half of the whole city, and had Christians in their houses as menservants and maidservants, who were open backsliders from the faith of Jesus Christ, and judaized with the Jews."
Part of Philip's shock is due to the landed wealth accumulated my members of the community, and partially due to the idea that they would have Christians as servants. The hierarchical structuring of medieval society made it difficult for Christians to conceive of non-Christians in positions of authority, and knighthood carried with it both status and a suggestion of wealth.
TLDR: while it is impossible to say that there was never a medieval Jewish man who claimed to be / was knight, prescriptive and narrative literature indicates that it would have been conceptually impossible. Jews were unable to participate in the explicitly Christian ritual of knighthood, and the idea of a Jewish person having authority or social status over Christians (even low ranking peasants and / or petty bourgeoisie [for lack of a better term]) was shocking.
Sources:
As far as I know, Jews were generally exempt from military duties in the period I study. They were forbidden to own land and were entirely confined to Jewish districts in towns and cities, so I can't imagine one possessing a knight's fee. In a real emergency, such as a siege, this prohibition may not have applied, but I can't say that definitively.
I do know that Henry II's Assize of Arms ordered Jews to surrender any mail coats they might have owned, so that they could be given to gentile fighters, but it doesn't discuss any other arms Jews may have owned. Given the frequency with which Jews were slaughtered by angry mobs of townsmen and peasants, I suspect they didn't have much in the way of weapons.
I really can't think of any scholarship that addresses this issue directly, but neither can I recall any primary sources that mention their participation in war. I apologize for being so brief, but it's the best I can do with the very limited sources available.
Previous answers have focused solely on the opportunities – or rather lack of such – for Jews in Latin Europe. That’s understandable, considering it’s the region that comes to mind first for most people when they hear the term ‘Middle Ages’ and the second question about Jewish knights obviously already lead in that direction. However, I want to emphasize that medieval Jews didn’t only live in societies that were dominated by Christians. Far from it! The religion had originated in the southern Levant and even after the destruction of the Second Temple in 70 AD there were still Jews living there. Large and hugely important diaspora communities existed since Antiquity in places like Mesopotamia or Egypt. Their trading activities lead Jewish merchants as far as Sub-Saharan Africa and even China. A religion could hardly have a more global reach before the Columbian Exchange.
So, the obvious question is: were there Jewish warriors outside of Christian Europe? Thankfully the answer is yes, and we don’t even have to leave the continent to find them (sorry, no Jewish-Chinese warrior monks from me). My first example is from what is today the kingdom of Spain. The Iberian Peninsula was home to a large and vibrant Jewish community that from the 8th century onward was mostly living under Islamic rule. Similar to the situation in Christian Europe Judaism was obviously not granted equal rank to the rulers’ religion of Islam. However the peninsula’s Jews did have opportunities that their northern brethren lacked and they prospered for it. Under the Umayyad Caliphate of the 10th century Jews could already reach high positions at court. That continued even after the Umayyad state had fractured in the 11th century and given rise to a whole host of successor kingdoms, the Taifas. This was a tumultuous time full of warfare and intrigue but also cultural efflorescence, a time in which a gifted individual could rise high in the world. Such a man was Samuel HaNagid (993-1056 AD). In the last years of Umayyad rule he had enjoyed an excellent education in Hebrew as well as Arabic at the imperial capital of Cordoba. But when civil war broke out and Cordoba was sacked he and is family became refugees. His education nevertheless payed off because it brought him to the attention of the Taifa-ruler of Granada, Habbus al-Muzaffar. Samuel was named tax-collector, later court secretary and then even vizier. In this role he even became influential enough to meddle in the succession struggles after the death of Habbus, managing to bring the later’s son Badis to the throne. This marked the zenith of his career. He was held in high regard well beyond the borders of Granada, which among other things is shown by his epithet ‘Nagid’, meaning ‘prince’ in Hebrew. I mention him mainly be-cause his responsibilities included not only accompanying his ruler into war but even to act as a general himself. A Jewish man leading Muslim armies into battle! As the learned man that he was Samuel recorded his experiences on the battlefield in a kind of poetic diary, written in the Hebrew language and Arabic meter. Let me give you an example:
Lion-like warriors were with us, and soldiers like locusts in swarms: | men weighed down, though not with flesh, not from the fat on their loins. | All in their wrath who’d bear armor, all who’d strip their robes in war, | and all who’d hasten toward the sword, all who’d run toward death with joy; | with coats of metal mesh like waves and shields reddened as parchment, | on horses like clouds rushing and flying, | with rounded helmets and greaves; | their hands not full of lances and spears, like tamarisk trees— | the flashing of swords alone could be seen— no more was heard than their shouts of joy! (Peter Cole, trans., Selected Poems of Shmuel HaNagid. Princeton 1996.)
Now, the life of Samuel HaNagid is obviously an exceptional story. It demonstrates that the social roles that Jews could inhabit under Islamic rule were not nearly as circumspect as in those of their brethren living in Christian polities. But of course Muslim Spain was no paradise of complete re-ligious tolerance. The fate of Samuel’s own son Yehoseph shows this quite clearly. Samuel’s staggering success had engendered his fare share of envy by his Muslim peers and after his death in 1056 AD Yehosef proved far less adept in fending off their enmity. In 1066 his palace in Gra-nada was stormed by a violent mob, who not only killed him but even started slaughtering a large number of the city’s Jewish inhabitants – one of the few actual pogroms in the history of Muslim Spain.
So, even under Islamic rule a Jewish warrior was an exceptional occurrence. To find a medieval society were this was not the case we have to jump from one end of Europe all the way to the other, into the steppes of southern Russia. Like Muslim Spain they were still beyond the reach of Christian rulers. They are part of a vast steppe corridor that crosses much of the Eurasian landmass from the Danube in Europe all the way to the borders of China. This landscape is best suited for a pastoral way of life and so since Antiquity it saw the rise and fall of powerful nomadic empires. Their elites often practised their own pagan religions but since the steppes served as routes of connection between the different societies of Eurasia they also came into contact with all kinds of other faiths. Sometimes it could be attractive to convert to such a foreign religion. This in turn opened those communities to influence from the settled societies which could be used as a stabilizing force for the nomadic empires. Some of them chose religions that were dominant in neigh-bouring powers, like the Volga Bulgars or the Karakhanids who converted to Islam, or the Alans who became Christian. Others chose to differentiate themselves from powerful neighbours by embracing more obscure faiths. For example the elites of the Uighur Khanate chose Manichaeism, a minority religion from distant Iran instead of the Buddhism, Confucianism and Daoism that were favoured by their rivals in Tang China. This brings us back to Southern Russia, where the elites of the Khazar Khanate were confronted with a similar choice.
The Khazars had established their power between the Black Sea and the Caspian Sea in the 7th cen-tury AD. Here they became involved in the military struggles between the Byzantine Empire and the Arabic Caliphate. These two powers also represented rivalling religions: Christianity and Islam. Since the Kazhars often acted as allies to Byzantium a conversion to Christianity might have been the obvious choice. However they proved reluctant to take this step. Instead the Khazars seem to have opted for a third option. At some time in the 8th or 9th century AD their elites, including the Khan himself, began to embrace Judaism. It is unclear how large a part of Khazar society actually became Jewish but it seems to be beyond doubt that the religion became an important part of their elite culture. For several centuries the Khazar Khanate was the dominant power north of the Caucasus. Its rulers stood in contact with Jewish communities across the known world and projected themselves as protectors of the Jewish faith. Mesopotamian Jews seem to have hoped for a Khazar victory over the Caliphate. However in the 10th century their power began to dwindle as they became squeezed between the emergent power of the Kievan Rus in the north west and new nomadic tribal coalitions in the east. After the fall of state evidence for Jewish life in the South Russian steppes is lacking. There seems to have been no continuity between the Khanate and the Late Medieval and Early Modern Jewish communities in Eastern Europe.
I hope that these two examples are enough to show that Medieval Judaism was far from restricted to regions under Latin Christian rule and that in some cases the Jewish experience could indeed include the life of a warrior, even if not that of a knight.