We don't know, but it was probably during the Great Depression.
I was surprised to find, when I began my research into this question, that the U.S. Census does not track the number of people — citizens or non-citizens — who emigrate from the United States. That means there's no central clearinghouse for this data, which makes tracking emigrations difficult. Furthermore, it's difficult to distinguish between alien residents and citizens' departures, and temporary versus permanent departures.
Nevertheless, we do have some studies of emigration from the United States. In 2006, the international labor office of the United Nations Secretariat took a look at this phenomenon in recent history. It analyzed the number of U.S.-born and U.S. citizens living abroad in different countries and compared those to previous counts to get an idea of how many people were leaving each year.
In the case of Canada, it compared figures from 1991 and 2001, finding that there were 28,985 more U.S.-born people in Canada in the latter year than the former. Using a raw average, that works out to a net outmigration of 2,899 U.S.-born people per year to Canada.
There's a lot of issues with that figure, and you can read an analysis in the link above if you're interested in how there really aren't 2,900 people per year moving from the U.S. to Canada. (Hint: It involves births.)
A 2009 study by the Congressional Budget Office looked at emigration rates among recent immigrants and found that between 1 percent and 1.5 percent of the foreign-born working population emigrates from the United States every year, a pattern that exists as far back as the late 1970s.
Since these two studies are within the 20-year verboten period for this subreddit, I won't talk about them.
Doing a bit of research, I did discover a small little article from the Aug. 18, 1932 issue of the New York Times, which declares that "this is the first year in the history of the country when the number of aliens permanently departing from the United States exceeded the arrivals."
According to figures from the U.S. government, 103,295 alien residents left the United States in fiscal year 1932 (ends June 30, 1932) while only 35,576 permanent immigrants were admitted. More than 32,000 of the departures were due to deportations, but that still means that almost twice as many resident aliens willingly left the United States than arrived.
Why? Well, columnist Harold Fields took a stab at that question in the April 1, 1934 New York Times. He comes up with a few reasons:
• Many of the people leaving are old, heading back to their homeland for sentimental reasons. • There's a depression, don'tcha know? • There were plenty of deportees mixed in. • Many of those leaving are peasants who have gone broke. • Their homelands want them back.
Even before the FY32 report, however, there were signs that people were leaving the United States because of the Great Depression. In 1930, Canada reported receiving 50,588 immigrants from the United States — almost 3,000 more people than left Canada for the U.S. The same year, emigration from the United States to Ireland exceeded emigration from Ireland to the United States — but by only 604 people.
Looking back farther than the Great Depression, there were plenty of attempts to get people from the United States to emigrate to new locations in the 19th century, and in the years before the Civil War, many of those attempts involved America's free black population. You're probably familiar with Liberia, which was founded by the American Colonization Society in 1821. Even by 1859, however, there were at most 12,000 American blacks in that territory, as compared to an estimated 500,000 across the United States.
Beyond Liberia, there were attempts to lure free blacks to places including Jamaica and Grenada, but none were very successful.
On the flip side, there were the machinations of William Walker, who invaded Nicaragua and attempted to set up a slaveholding nation that would welcome settlers from the American South and be a loyal slaveocracy in Central America. Needless to say, he was not successful.
In short, even the largest known emigration from the United States — that during the Great Depression — was not very large when compared to the nation as a whole and did not even last through the whole Depression.