Maybe New Spaniards, but not Mexicans. I believe Mexico would have been called New Spain before 1776.
Make that 1810 - the date when Mexico became independent of Spain.
The CIA World Fact Book (which 0bama must not have read) has the following entry for Mexico:
16 September 1810 (declared); 27 September 1821 (recognized by Spain)
In 1813, their declaration of independence was issued, under the name of America Septentrional or "Northern America" - in the sense of being the northern portion of Spain's former American possessions.
After years of war, Spain and America Septentrional signed a peace treaty in which the country was designated as the "Mexican Empire."
After two years of imperial rule, it became "The United Mexican States".
In 1836 the name was changed to the Republic of Mexico, and it has since been changed back to "The United Mexican States."
A better case can be made that there were Spanish-speaking settlements in parts of what is now the USA, before the USA became a nation, just as there were some Franch-speaking and some Russian-speaking settlements. But small and few. The US southwest was far more Indian than Spanish. The vast, vast majority of Spanish-speaking New Spain colonists lived way south, south of the Sonoran Desert, down in central and southern Mexico.
Even the Spaniards weren’t here in any number. Just a few small isolated settlements like Santa Fe and Los Angeles. The Commanche and Apache ruled north of the Rio Grande and they were a formidable barrier to any Mexican and Spanish presence. It took the US Second Cavalry years to subdue the Apache and Commanche. Mexicans were no match for them.