Yes, and in staggering numbers.

You ask about mercenaries, and let me be clear: Neither the United States nor Confederate States went out to hire foreign mercenaries to fight in the American Civil War on a large scale. There are, however, ample instances in which both the U.S. and C.S. hired foreign soldiers and sailors who then fought under their respective flags. There is a distinction.

The Confederate Navy was perhaps the most notorious force in this regard. Charles M. Robinson's Shark of the Confederacy: The Story of the CSS Alabama explains how Confederate Secretary of the Navy Stephen Mallory faced the titanic task of building a navy in a country that had little shipbuilding capacity, few sailors and even fewer naval officers. After President Jefferson Davis rejected Mallory's request to buy ships ready for combat, Mallory switched to raider planning.

The most famous Confederate raiders were built in Great Britain, and as such the Confederacy faced huge challenges in manning and equipping these new-built ships. Because of neutrality law, these ships had to be armed and crewed in such a way that Britain could not be held responsible. (Indeed, in 1863, the Laird Rams affair brought the United States and Britain closer to war than at any time since the Trent Affair.)

Gabriel Raphael Semmes, captain of the Confederate raider Alabama, built in 1862, was extraordinarily aggressive in recruiting foreign sailors to serve aboard the Alabama. "Their reasons for joining up were varied," Joseph McKenna writes in British Ships in the Confederate Navy, "financial gain not the least. Many were in the Naval Reserve, former Royal Navy men who had served during the late Crimean War. They were experienced sailors, battle-hardened, and not content to finish their days aboard some rust-bucket in coastal waters around Britain. They sought adventure, and in the service of the Confederate Navy, they found it."

I believe these recruited sailors are the closest thing to what you're asking about.

Those sailors were not, however, present in staggering numbers.

In the armies of the North and South were tens of thousands of men, new immigrants to the United States. Between 1820 and 1860, almost 4 million people immigrated to the United States, and many of them were from Europe, where antislavery sentiment was far stronger than it was in the United States. Of the approximately 2 million soldiers who enlisted for the Union during the war, almost one-third were not born in the United States. These included about 200,000 Germans, 150,000 Irish and 150,000 from British territories. This article by a re-enactor explains things in plain English with some citations, but if you have some questions about the source (as I do — I just prefer its plain-English tone), I'll continue.

The most famous of the foreign-born units is the Fighting Irish Brigade, formed under the influence of Thomas F. Meagher. Altogether, the Irish dominated at least 20 regiments in the Union Army. Their fame somewhat overshadows the participation of other foreign-born units, even though German, Hungarian and others made up a greater percentage of the Union Army.

Germans are of particular note. The failed 1848 revolution in Germany forced many liberal Germans out of Europe and to the United States. They were anti-slavery almost to a man (and woman), participating in the fighting almost from the first day. In May 1861, with a showdown between pro-Southerners and pro-Northerners brewing in St. Louis, the huge German community was key in keeping St. Louis in the hands of the Union Army and driving the Confederacy into the Missouri wilds.

U.S. Gen. Franz Sigel, a former German military officer, became famous for issuing orders in German, having them translated into Hungarian for his officers (many of whom had fled that nascent country) and then into English and German again for his soldiers.

My favorite German of the war, however, is Carl Schurz, who as Ted Widmer wrote in 2011, was "A 19th century Zelig of sorts[.] [H]e had a knack for arriving just as trouble was erupting, and for leaving just as he was about to get caught."

Schurz was a 19-year-old student at Bonn University when the 1848 revolution struck. He was an ardent follower of the liberal revolutionaries, and when his teacher, Gottfried Kinkel, was thrown into Spandau prison for his political views, Schurz led a Mission Impossible-style rescue. "The escape was a sensation and made Schurz the most famous teenager in the world," Widmer wrote.

After the revolution failed, Schurz spent a few years in England, got married, had kids, and then in 1852 came to America, where he opened the nation's first kindergarten. He joined the Republican Party almost as soon as it got started, and Lincoln rewarded him for his support by naming him ambassador to Spain — a shocking move, considering he was a revolutionary now in an aristocratic stronghold.

In 1862, with the war ranging, Schurz returned to America to fight in the war. He was appointed a brigadier general and would go on to command a division, fighting at the Second Battle of Bull Run, Chancellorsville, Gettysburg and elsewhere. He survived the war, won election to the U.S. Senate and later became Secretary of the Interior under Rutherford B. Hayes. He died in 1906, and in 1917, the U.S. Navy renamed a captured German ship the U.S.S. Carl Schurz.

Germans and Irish weren't the only volunteers. There were participants from every country in Europe, almost. There was the 55th New York Infantry, for example. That regiment's original recruits came from French émigrés in New York City. Even by January 1862, six of the regiment's nine companies were predominantly French. As James Johnston wrote in 2012, "Some of the men were veterans, having served in the French Army in Algeria, the Crimean War and Italy. The rest were a motley international bunch, including German, Irish, Italian and Spanish immigrants, as well as a few Americans."

The Confederacy, with fewer immigrants, had fewer foreign fighters within its ranks, but it had them. The 10th Louisiana is one such unit. As Terry Jones wrote in 2012 of its commander, Col. Eugene Waggaman:

"Waggaman was a deeply religious and popular officer who commanded a motley crew of soldiers. Most of his men were from New Orleans, and they reflected that city’s cosmopolitan makeup: one company had recruits from 15 foreign countries and another was made up almost entirely of Greeks and Italians. Because many of Waggaman’s men could not speak English, the regiment used French drill commands exclusively. It was motley in other ways as well: the men of the 10th Louisiana were constant camp discipline problems and helped create the notorious reputation of the famed and feared Louisiana Tigers. It would also see its share of hard service: of the regiment’s 845 members, 205 would not survive the Civil War."

You'll note that I've used the word "foreign" throughout this answer, but after writing all this, I think I have to change my answer. They might have come from countries around the world, but in the end, they fought under two flags. In the end, they were all Americans.

Mexico was busy fighting France. The Cinco de Mayo holiday originated around the same time, after the Battle of Puebla, and even spread to camps in the states.

Canada was still British. The South begged Britain to join their cause, and they thought that the British would want to break the blockade for access to American cotton. The general feeling in the UK was anti-slavery (they abolished slavery 30+ years before us), and the British had access to other cotton rich areas in Egypt and India, so they didn't need the south. For the most part, Britain stayed out of the conflict, and so did Canada. FWIW, Canada was granted Dominion status only 2 years after the conclusion of the Civil War, thought I'm not sure if the two events were in any way related.

I can't speak to the possibility of mercenaries or foreign volunteers, though I doubt there were any Mexicans, since they were busy fighting their own war.

Can be argued Canada became a country because of the Civil War. I've argued this before, and it's a common one among historians. Copy pasta from earlier posts of mine.

The Union may have attempted to take over Canada to gain more land. The Civil War was one of the main reasons that Canada was allowed to have confederation so easily.

John A. and the other founding fathers believed that the Civil War in the United States was caused because of the 'excessive power given to the USA under the constitution.' [Source] The St. Albans Raid also helped to fuel the fears of the Union. The St. Albans Raid was confederates based in Canada, launching attacks in Vermont. After the attacks they were arrested, but promptly released by the Montreal Police. [Source]

Anyway, all of that puts everything in context. During the Civil War the Union had wrote the Annexation Bill of 1866 which called for the annexation of British North America. The reason the Annexation Bill was written was because of the Fenian Raids (Irish Raids into Canada), as well as a common feeling of resentment among the Union population because of the British support for the Confederates during the Civil War. [Source + Annexation Bill 1866]

The source I have provided a link to above. This was shortly after the Civil War had ended. To me, this shows that this had been a plan that had been considered for a number of years, possibly all throughout the war, as a result to the hostility that the Union had towards Great Britain. The way it has been explained to me, was that if the Confederates had been succesful with the seccession then the Union would move to Annex Canada. The reason the Union would do this is because of Britains active support for the Confederacy all throughout the Civil War. In 1861 The Trent Affair lead to 11,000 British troops being sent to the Canadian border. There was a real fear that the Union was going to try and attack Canada.

So, one can make a fairly legitimate argument showing Canada became 'Canada,' because of the Civil War.

Oh, I also don't know about any volunteers (that's not to say there weren't any). Military history is but a faint interest of mine. I was just merely speculating on the political/social impacts of the Civil War.

The American Civil War had a particularly insidious result for Mexico. Mexico had just finished up her own civil war (the Reform War) when the US Civil War broke out.

The some of the Conservative losers of the Reform War fled to Europe where they successfully convinced the French Emperor Napoleon III to invade Mexico and establish a Mexican Empire with a European prince as ruler (Maximilian of the Hapsburg House).

Key to Napoleon III's decision to invade Mexico, was the belief that the US would be too distracted with her Civil War to intervene in the establishment of the Second Mexican Empire.

With the end of the American Civil War, the US was again in a position to intervene against the French, and this proved one of the reasons for the withdrawal of French troops from Mexico in 1866.

Source: Napoleon III and His Carnival Empire by John Bierman

Liberia was extended diplomatic recognition by Congress in February 1862. The election of Lincoln and the departure from Congress of most southern Representatives and Senators left the Republicans free to pursue a long standing abolitionist goal of recognition of Liberia (and also Haiti). I'm not familiar with any official stance on the Civil War from Liberia's government itself but I know they had been seeking recognition from the United States since they declared independence in 1848.

Even before the French intervention in Mexico (starting December 8, 1861) Benito Juarez was a firm supporter of the Union though some local officials such as the governor of Nuevo Leon y Coahuila enjoyed warm relations with the Confederates. A big driver of Juarez' support for the Union was a recognition that the interests of the American slaveholding class was the driving force for territorial expansion into Mexico and that Lincoln was an opponent of the US-Mexican War. Juarez even had the Confederate commissioner John Pickett arrested and expelled in 1861. When the French intervened in Mexico they were discreetly supportive of the Confederacy but both the French and Emperor Maximilian abstained from recognition and overt support of the Confederacy.

Sources: James Oakes, Freedom National: The Destruction of Slavery in the United States, 1861-1865 The Struggle for the Recognition of Haiti and Liberia as Independent Republics https://archive.org/stream/jstor-2713395/2713395_djvu.txt Howard Jones, Blue and Gray Diplomacy: A History of Union and Confederate Foreign Relations Robert E. May, Manifest Destiny's Underworld: Filibustering in Antebellum America