'Nuff said. They are still an independent country in 1860, and definitely don't want to give up slavery. How does history play out?
1) No Mexican/American War. Mexico retains title to much of the American Southwest. California possibly revolts at some time in the mid-1850s and secures it's own independence, especially if gold is discovered on schedule in 1849 and there is a huge influx of Americans and other foreigners into the territory.
2) Possibly an independent Mormon nation of Deseret. The land where the Mormons settled in Utah would have been in Mexico. If they chose to rebel at some point, they were probably too far away from Mexico for the Mexicans to hold onto them.
3) Quite possibly no American Civil War in the 1860s, as there won't be as many controversies over the admission of additional western territories to completely polarize the two sides against each other. The Missouri Compromise remains intact, slavery eventually dies a natural death sometime between 1890 and 1910.
4) Alternatively, there is still a Civil War, but Texas remains neutral and trades with both sides. The Union is MUCH weaker in this scenario because it doesn't have the California and other western gold and silver reserves to back up it's currency and purchasing power abroad (the Union did import huge quantities of arms and equipment during the war...more so than the Confederacy, in fact. It is true that it also produced most of what it used during the war, but if imports are harder to get, or if sound money to pay for purchase contracts doesn't exist, it's ability to build and equip armies is going to be significantly impacted). The Confederacy, despite the loss of Texas manpower, is actually somewhat stronger than in OTL, because the ability of the Confederacy to use Texas ports to evade the blockade makes it easier for it to bring in the supplies it needs to equip it's own armies.