Personally, I think it's more important to potential employers that you end up with your BSN vs which school you went to. If you are going to a community college program for your RN, then you can get your BSN in a little over a year after your initial RN.
If you get your initial RN from say Mesa CC, then go to ASU, GCU, UA, or NAU for your BSN... your BSN is from that school (if names really matter to you).
Let me put it to you this way... In all of the applications I have filled out for RN jobs, everyone of them asked me if I had a BSN or not. This didn't used to matter as much... but today every little detail to get your application to a hiring manager's desk is important.
Good luck in your program!