Answer:
They were fighting their own civil war at that time. The French emperor Napoleon lll had taken advantage of the fact that the US was distracted by their Civil War and would not able to enforce the Monroe Doctrine which was America's promise to protect any country in the Western hemisphere from European colonization. Given the chance, Napoleon lll installed a Hapsburg along with 40,000 French soldiers to occupy Mexico and enforce his regime. The new government spent it's entire existent trying to subjugate the country and therefore couldn't have played any role in the American Civil War. They were busy fighting a war of their own.
Step-by-step explanation:
So generally, 1861 was a interesting year for both nations (Mexico and the USA) , while one just finished it's Civil War, the other started it. Mexico established a federal republic while America was about to abolish their institution of slavery.
Mexico was in pretty bad shape and I don't doubt that private Mexicans fought for either side. The military involvement was minimal.
But as the saying goes: "the disgrace of one man, is the happiness of the other"
The Confederacy failed to get recognized as a nation by any European power, especially by the United Kingdom, although, it traded tobacco and cotton with Dixie, the British wouldn't do it because it would seem hypocritical to defend a anachronistic institution when you abolished it in 1807.
With Scott's Anaconda Plan, the Union imposed an embargo. The C.S.A wasn't able to use it's ports.
Without Europe's help, and the U.S fighting in the North, they would have to look South of the Grande River, so they did. The C.S.A sent Cuban-American, Jose Agustin Quintero to establish diplomatic relationships with Santiago Vidaurri, Nuevo Leon-Coahuila governor or "Caudillo del Noreste" (Northeastern chief) to let them use Tamaulipas' ports (especially Matamoros) to send their products overseas.
The deal was a win-win, because with the opening of these waterways, both economies would benefit, the South's by getting to sell it's produce without being harassed by the U.S Navy, and the Northeastern Mexico (Nuevo Leon, Coahuila and Tamaulipas) with the tariffs that recolected from those shipment.
These situation would be uncomfortable to Benito Juarez, given that he was sympathetic to Lincoln's efforts.
Possible Alternative
How was New Mexico divided at the beginning of the Civil War?
- By seizing the southern portion of the Union's New Mexico Territory, the Confederacy founded Arizona, and the line drawn in 1863 created an Arizona on the west that was divided from New Mexico on the east.
Thank you,
Eddie E.