"Stake a claim to" is most common, as shown by this Ngram. About twice as common as "on", four times as common as "in".
If one goes back to 1920, however, "stake a claim on" was more popular. This likely related to the fact that it was more literally understood back then -- a process of driving stakes into the ground around a parcel of land.
From the standpoint of syntactic or semantic validity, though, all formations are equally valid. So a writer is free to choose whichever seems to best express the desired meaning.
In the original examples I'd likely say
Many homesteaders moved west to stake their claims in the Nebraska territory.
Hundreds of thousands of Mexicans immigrate to the U.S. every year, seeking to stake their claims to American soil and lead a better life.
(Though that second one is a difficult call.)