The most famous story of North American colonialism is obviously the Mayflower Pilgrims, who emigrated to America in order to avoid loss of religious/cultural identity. This seems like a problem which European Jews would likely commiserate with during the period of American colonialism. Were there any Jewish Mayflowers?
I think that the best example of this in the colonial period is the Marranos who left Spanish-controlled lands in order to return to Judaism. In 1391 was the first massive wave of coerced conversion of Spanish Jews, and in 1483 the Spanish instituted the Inquisition in order to police these New Christians. By 1492 the Inquisition had successfully convinced Ferdinand and Isabella to kick all practicing (non-converted) Jews out of Spain. In a sense, that could count as an emigration to escape religious persecution- Jews generally went west to Portugal, south to North Africa, and east to the Ottoman Empire. In 1497, the king of Portugal forcibly converted all Jews in the country, and by 1536 the Inquisition established itself there as well. Spain and Portugal (soon to become one country under the Iberian Union). Thousands of New Christians remained on the Iberian Peninsula.
Many of these New Christians were completely Christian as far as their religious theology and conduct. However, due to the Spanish policy of limpieza de sangre (blood purity), these New Christians were limited by their Jewish ancestry in terms of their social status and ability to socially climb. This was the first time that converts were given a racial distinction. Previously, once a Jew converted (and the Church worked HARD to get Jews to convert), s/he was as Christian as any other. Now, there was a social barrier created by one's ancestry. People had to have family pedigrees in order to prove that they were Old Christian- of course, forgeries abounded which masked New Christian ancestry.
Of course, right around the same time as the Expulsion was Columbus's expedition to the New World (in fact they were within a few days of each other), and the New World ended up providing not only a source of wealth but a way for many to start anew. Many of these New Christians who were Christian at heart but resented being judged by their or their ancestors' former Judaism often traveled to the New World, where they could make new names for themselves and improve their situations.
With those New Christians, often called Marranos, who continued to secretly observe Jewish practice, many of them traveled throughout the world, including to the communities of the Spanish-Jewish post-expulsion diaspora, in order to return to Judaism in the several hundred years after the expulsion. By the 1600s, this had expanded past the areas mentioned above and a very large community had been established in Amsterdam, which became renowned as a center of Sefardic Jewish life and trade. However, many also traveled to the New World, where, conveniently, an Inquisition was not yet established (it wouldn't be until 1571) and it was easier to stay under the radar. Even after the Inquisition was established, there were still clandestine communities of Marranos who observed Judaism- the most famous is probably the family of Luis de Carvajal el mozo/Joseph Lumbroso, whose story is absolutely fascinating.
Almost all of Central and South America were controlled by the Spanish/Portuguese and ended up with an Inquisition at some point, but in 1631 Recife/Pernambuco, an area on the coast of Brazil, was captured by the Dutch and soon became known as a haven for Jews, whose population at some points purportedly was twice that of the Christians. Many Marranos either lived as Marranos in peace there or returned to normative Judaism. However, Recife was recaptured by the Portuguese in 1648 and that period of Jewish utopia ended. Jews also settled in areas like Suriname (which had a city called Jodensavanne, or Jewish savannah) and Jamaica, which were controlled by the British, and Curacao and Sint Maarten under Dutch control. Of course, in 1654 when the Dutch took control of New York, Spanish Jews (some of whom were escapees from Recife) became some of its first settlers.
There is a good book called "Jewish Pirates of the Caribbean" that details early American Jewish history.
A lot of Jews came to the Americas to escape Spanish/Portuguese persecution, and when South America was no longer safe, they fled to Dutch-held New York.
Jews have been emigrating to the Americas since the beginning, but these were often small numbers. Mass immigration from Eastern Europe only really started in the later 19th century (when my own ancestors made the journey)