It depends on what you mean by "full rights." If you're asking when people were finally treated equally under the laws and structures of the United States government, that STILL hasn't happened yet. Remember, just because something is law doesn't mean it is enforced everywhere, or that people believe in the validity of the law, or that people are willing to follow the law.

While the 13th amendment might have freed the slaves on paper, it didn't erase the prejudice, fear, and racial assumptions of many whites in power (while it's easy to say that these were "Southern" issues, the truth is many northern whites held equally racist assumptions - it was easier to be anti-slavery in theory and in speech than it was in practice or when you were going to be competing with these newly freed individuals for jobs, land, and materials). After the 13th amendment many former slaves, bereft of substantial money or property or political clout, were forced to work at or under subsistence levels through sharecropping. Some states and communities established "Black Codes" - laws modeled after slave codes - to attempt to control the black population. Even after these were outlawed, literacy tests, voting taxes, and lynching continued to attempt to police black attempts at gaining and sort of real power (or even attempting to merely survive). These forms of oppression may not be present today, but systems still exist that operate to marginalize blacks and people of color - which is why the "rights" question is so hard.

As for other groups - whiteness in America has often been defined by "othering" those that looked or acted in a way that was opposite of what established whites wanted to be seen as. Again, this still exists today - immigrants as criminals, eating strange foods, able to subsist on small amounts of money and materials, hyper-sexualized, prone to violence...the list goes on. By "othering" these people, American whites hope(d) to establish whiteness as being the opposite - moral, peaceful, proper, needing fine comforts and fine dining, "good" citizens.

Again - depending on how you define rights, you can make the argument that this still presents a dismal level of inequality.

As to your question of whether Mexicans, Chinese, or other non-white laborers could be land owners - well, that depends on where and when you're talking about. In some places, freed slaves could own property, while in others entire populations of non-whites were either driven out or exterminated in response to fears of job competition or fears of racialized sexuality/violence.

ALSO: Some people try to make the argument that Irish indentured servitude in the early years of the U.S. is an example of white slavery which therefore invalidates the argument that blacks and people of color have faced, and continue to face, an uphill struggle. While Irish and other poor whites were at times forced into unpaid labor, the comparison falls apart almost immediately - they were nowhere near used as long, or in the same number, they often were indentured for a set number of years, rather than generations, they were generally not treated as poorly or inhumanely, and most importantly, their marginalization and dehumanization did not become a cornerstone of U.S. policy, ethics, and attitudes as it did for blacks.