I've never heard of anything like this. Borders in the nineteenth century were very porous so one wouldn't necessarily expect everyone who went to Canada to never cross again, and black Canadians could migrate for all the same reasons anybody else might, but the claim that black people as a group required an escape route from Canada to the United States doesn't make sense unless they, as a group, have a good reason to flee Canada and to expect the United States to be significantly better. I'm aware of neither and intensely skeptical of the latter, though as I'm no Canadianist I'm happy to be corrected on the matter.

The closest I can come involves slaves who escaped during the War of 1812, when the Royal Navy engaged in fairly regular raiding around the Chesapeake, and later points further south, and often found people, mostly young men, quite eager to hitch a ride. Often they would come back later, dressed in British uniforms and with friends of similar fashion sense, to get their families. About five thousand American slaves escaped during the war, 4,192 of whom we know the British resettled 2,811 in Nova Scotia and 381 in New Brunswick. The British previously relocated about two thousand to Nova Scotia at the close of the War for Independence, so it makes good sense that they'd see it as the best place to use a second time over.

There was a theory advanced at the time, and still sometimes repeated, that the perfidious British seduced slaves away from their happy homes and family, taking them far away on a tissue of lies and then selling them into harsher slavery in the West Indies. The British investigated the charge seriously and found no evidence for any of it. Nor have subsequent historians.

Still, some Americans took it very seriously. John Quincy Adams, in Europe to negotiate the Treaty of Ghent, pressed the British on the issue. He thought that proof existed back home. It did not and a couple of years later he got to write James Monroe and explain that this was all make-believe.

Some of these freedpeople did quite well in Canada. At least one wrote to his old owner to brag about his success:

Sir, I take this opportunity of writing these lines to inform you how I am situated hear. I have [a] Shop & Set of Tools of my own and am doing very well when I was with you [you] treated me very ill and for that reason i take the liberty of informing you that i am doing as well as you if not beter. When i was with you I worked very hard and you neither g[ ave] me money nor any satisfaction but sin[ ce] I have been hear I am able to make Gold and Silver as well as you. The night that Cokely Stoped me he was very Strong but I showed him that subtilty Was far preferable to strength and brought away others with me who thank God are all doing well. So I Remain, Bartlet Shanklyn P.S. My love to all my friends. I hope they are doing well

Taylor, Alan. The Internal Enemy: Slavery and War in Virginia, 1772-1832: Slavery and War in Virginia, 1772–1832 (Kindle Locations 109-114). W. W. Norton & Company. Kindle Edition.

Not everyone was happy with the British, though. Nine slaves did end up coming back, young men who fled quite late in the war and so didn't have as much time as others to adjust to their new lives. One didn't have any choice in departing; the British just took him. None of them fled with their families. None had made it to Canada. All appear to have had bad experiences while in British service. Were there more that just didn't make it home?

Maybe, but the Monroe administration was keen to get American property in people back and dispatched special agents to talk to the resettled slaves in Bermuda and Halifax. In Bermuda, Thomas Spalding demanded that the British governor just round up the freedpeople and ship them back by force. He declined, and it seems chewed Spalding out pretty good, but did permit the Georgian planter to go talk to the ex-slaves. (Incidentally, this governor, James Cockburn, was the brother of a Rear Admiral Cockburn whom Spalding had tried to get to leave runaway slaves behind when he evacuated Cumberland Island, Georgia, a few months prior.)

Spalding gave it a go. In a move sure to surprise only an enslaver, none of the ex-slaves thought it would be cool to go back to slavery. Spalding couldn't accept that and claimed that the British had poisoned the slaves against Americans. They did so, of course, entirely so they would have the slaves on hand to use against the United States in a future war.

Spalding also squabbled with a second agent, Eli Magruder. He tried the secret agent approach, thinking he might recover the slaves through the courts. The Bermudian lawyer he consulted on the case? Attorney general for the colony. Said official told him that absent the governor's order, nobody was leaving Bermuda against their will. Magruder also thought he was in charge and tried to boss Spalding around, which ended with Spalding quitting Bermuda.

Right, but that's Bermuda and the question is about Canada. That was Augustine Neale's assignment. He was supposed to go to Halifax incognito. He was to find evidence that the British sold runaways back into slavery and generally destroy the impression of Britons as liberators of slaves in order to consolidate slavery back home. To woo the freedpeople back, Neale hired one of the nine who had returned as a kind of character witness. In order to prove that Neale was totally, 100% on a private mission and not working for the United States government at all, Monroe dispatched him to Nova Scotia on a Navy ship. The New York papers reported on Neale's mission before it sailed. And its crew were chiefly free blacks who, as you might imagine, were not on board with keeping the secret.

Neale arrived to a decidedly frosty welcome. There was plenty of racism in Nova Scotia, but white Nova Scotians ended up disliking white Americans more than blacks. Neale himself admitted that the ex-slaves had it better in Canada. His sales pitch to the freedpeople focused on how cold Canada was compared to Virginia and how they still had family back in the US. But the same family ties could keep them in Canada, since many had gotten away with some loved ones. And everyone knew they could expect punishment and probably sale if they went back.

Think Neale has it hard already? It gets worse. The captain of the ship that brought him, the Nonsuch, decided that this was the best time to enforce some strict discipline on his mostly free black crew. He imprisoned two. A third freed them and all three took off into Halifax. They spread further news of just what Neale was on about and soon another pair of black seamen joined them in town. They liked to go out to the wharf and wave at the ship. Presumably they used other gestures too, as Taylor includes reference to "insults".

Captain Trant had enough of that. He got a local constable to help him out and tried to haul his sailors back by force. A white mob intervened, knocking him down and kicking him. The returned slave that Neale hired to use as a character witness, Joseph Webster, came close to being killed in the affray. The Halifax courts told Neale that nobody was coming with him; he'd best stick to the ship and get gone.

This isn't encouraging, but Neale got one old woman to sign up for a return of her own, Harriot Johnson. The Nova Scotian government refused her a passport and ordered the Nonsuch to depart. It left without her and short about half its crew, which made for a net loss and a less than pleasant return voyage.

Let's leave aside incompetent secret agents for a moment, though. Some Georgians made an agreement with Halifax locals to do the wooing. Nobody bit, except some quite old slaves that they didn't deem worth the cost of returning. If anybody really hated Nova Scotia that much, they still had better prospects if they moved on to Boston or New York. Some of them probably took that option, as Nova Scotia wasn't exactly prime land to get a new start and the colonial government was ill-equipped to provide for them. The white inhabitants hadn't been hospitable to the last refugees after the Revolution either, prompting most to move along to either northern US ports or Sierra Leone. The usual ugly treatment of refugees ensued all over again, complicated by how relocation and promises came from Royal Navy commanders but then colonial governors were left to fulfill them.

That's about the closest I can come to a general movement away from Canada in favor of the US. You can say it's kind of what a reverse Underground Railroad implies, but I think that would be radically exaggerate the case. Freedpeople were surely mistreated and we ought not minimize that, but nothing on the scale of slavery itself (which had withered away in Nova Scotia in the decade prior) and so were hardly compelled to quit Canada for the United States in anything similar to the way they fled the Antebellum South.

Sources:

Alan Taylor's The Internal Enemy talks quite a bit about black resistance, self-emancipation, and flight to British lines during the War of 1812, including giving the numbers I've used here.