I don't have any information on a Maximo Aguirre, but both Anthony and La Union are in Dona Ana County, New Mexico, and were surely only recently not part of Mexico and of course are next door to Chihuahua state.
I noticed a number of Aguirre families in the 1860 and 1870 Census records. I recorded those from my line of Aguirres.
From "Free Inhabitants in Dona Ana in the County of Dona Ana, State of New Mexico, enumerated by me on the 26th day of July 1860, G. McFrazier, Post Office Mesilla, New Mexico
Mariano Aguirre, aged 64, male, Occupation Miner, Value of Real Estate, 4,000, born in Mexico
Severiana, aged 30, female, born in Mexico
Anacleto, aged 16, male, born in Mexico
Tomas, aged 14, male, born in Mexico
Modesto, aged 5, male, born in Mexico
Teresa, aged 12, female, born in Mexico
Sabina, aged 5, female, born in Mexico
Petra Griego, 25, female, Servant, born in Mexico
Agapito Gonzales, 35, male, Teamster, born in Mexico
1870 Census, Las Cruces, New Mexico
Domicile # 315,
Aguirre, Severiana, aged 44, Female, Keeping House, born in Mexico
--, Anacleto, aged 26, Male, Wagon Master, b. Mexico
--, Modesto, aged 17, Male, Laborer, b. New Mexico
--, Sabinita, aged 14, Female, At Home, b. New Mexico.
I have some information on the family of Mariano Aguirre as well, if you care to know.