I have lived with my grandmother in Texas for over 10 years. There have been times when I have tried to move out, but she wants me to stay. At this point of my life I'm fine staying here with her, I'm kind of a security blanket for her. One of my aunts is trying to get me to move out. Does she have the right to "evict" me? I think she fears that if I'm still living here when my grandmother dies I will try to keep the house, which I'm not. Do I have any legal rights since I have lived here for so long and do I have to move out if my grandmother, who is still making her own decisions, doesn't want me to leave? Can my aunt force my granny to kick me out?

Yes, since you have established residency, you are considered a tenant in the house. Even though you may not have a formal lease, or pay rental, you are still considered a tenant. To get you out the owner of the house would have to go through formal eviction proceedings.

If your aunt is not the owner, she would have to have unlimited power of attorney for your grandmother to be able to evict you.

If granny is still making the decisions, why not ask her to prepare a lease that gives you the right to live in the house as long as she does. She could word it that, once she permanetly vacates the house, you will have 60 or 90 days to find new lodgings. This may satisfy the aunt to leave you alone.

From my understanding sons and daughters basically inherit 'first' and your grandmother would have to put anything you get in her will. So your aunt's fears are unfounded that you living there could take possession when she dies,
As long as your grandmother is competent and sane then she should be the one to have the say. Your aunt shouldn't be able to legally evict you against her say.

What does your grandmother have to say about her acting like this?
Can you have a talk with your aunt and reassure her that you aren't trying to get anything?