I have asked my friend (not extremely close) who is losing her house to stay with me while she gets on her feet. What rights does she/I have?

What rights did you give her or what rental agreement did you have them sign.

But when they move in, they become a non paying renter ( unless you charge them rent) so they will have to evicted if they will not just leave.

Yikes, not close? I'd definitely have a written agreement. One idea is an escalating amount, such as 50 for the first month and add 50 each month (it hits 600 in a year but the average/mo is only 350). She will have full knowledge, and you will have recourse if you have to evict for nonpayment.