What universities (R1) require that their students own a computer?

depends which college you are looking for I know that University of
Virginia and Virginia Tech require that you purchase a pc from them
(this don't mean bring your own.. you have to buy thier pc at their
price)  I believe this is going for the norm for certain classes, I
went to University of North Carolina at Greensborro, and wasn't
required to have one (computer science major) so your best bet is to
call around to the colleges you are interested in and find out then

Bridgewater, Framingham and Worcester state colleges in Massachusetts
require students to own laptops. The University of Texas requires
students to own Apple laptops. Drew University in NJ and Wake Forest
University in NC include laptops in their tuition. Reportedly, Drexel
and Dartmouth also require students to own computers. I went to
Caltech, which did not have any computer ownership requirements, so it
all depends on the school. Your best bet is to ask the colleges
themselves.

If you don't own a computer these days, you are either opting for
manual labor or housewife as your target job.  (Okay, I'm joking, but
every industry seems to be headed that way).

The issue becomes more about whether you have to buy a computer from a
university - regardless of whether you already have one or not - OR
you have to run a specific OS to be allowed onto the campus network. 
Most universities supply a computer lab for those who don't own a
computer/can't afford one and simply encourage everyone to get a
computer.  Those that enforce a policy of a specific laptop or
whatever are doing it because they have to manage more than 1,000
people owning computers and a standard, consistent setup makes it easy
to push security updates and system policy changes to everyone.

If you go to a college that doesn't enforce buying their laptop, you
should consider the Tablet PC and get Microsoft OneNote to take notes
with.  I wish I had that when I was in college - would have been the
only thing I would have needed in class (assuming I could have gotten
PDF versions of the textbooks).