When Texas got independent itself, were native mexican  kicked out from their home to narrowed Mexico?

No. Neither were they kicked out when Texas joined the US. There are a lot of Mexicans in Texas whose families have been in Texas since before there was a United States.

All the Americans cared about when they created independent Texas was that they could take their slaves to Texas. They did not care one bit about the Mexicans. Otherwise, they would have petitioned to join the US right away. But the US would not allow any more slave states to join the union.

Mexicans in Texas did not like Mexico calling the shots for them or taxation without representation. Of course they also resisted it from the north. I imagine a great deal of the rebel nature belongs to natives of Texas, Mexicans. 

A lot of them stayed put & became texans & later Americans.

As Eva Longoria once answered an interviewers question on her family's nationality "We didnt crooss the border. We were here first & the border crossed us."