List of the universities using E-Verify campus-wide provided by Pam Beemer, NU Vice-President, Chief Human Resources Officer, along with links to relevant state laws for claims on Timeline (May 20, 2018). 

1.  Duke University NC - MANDATORY
All state agencies, offices, and universities must use E-Verify for new employees hired on or after Jan. 1, 2007, except for employees of local education agencies hired on or after March 1, 2007.
https://www.lawlogix.com/e-verify-map/north-carolina/

2.  Emory University GA - MANDATORY
January 1, 2012
https://www.lawlogix.com/e-verify-map/georgia/

3.  Ohio State University -  OH - No state E-Verify law

4.  Pennsylvania State University (state university) -  PA - State law obligates E-Verify for public work.
"Public work” is defined as “construction, reconstruction, demolition, alteration and/or repair work other than maintenance work, done under contract and paid for in whole or in part out of the funds of a public body where the estimated cost of the total project is in excess of $25,000, but shall not include work performed under a rehabilitation or manpower training program” (emphasis added).
https://www.lawlogix.com/e-verify-map/pennsylvania/ 

5.  Purdue University -  IN - State law limits deductions of employee wages from state income tax if not enrolled in E-Verify.
"Private employers are not required to use E-verify but will not be able to deduct employee wages from their state income taxes if they do not enroll."
University non- education-related employment could be affected.
https://www.lawlogics.com/e-verify-map/indiana/

6.  University of Arizona - AZ - MANDATORY
December 31, 2007
https://www.lawlogix.com/e-verify-map/arizona/

7.  University of Florida (public university) - FL - MANDATORY for all agencies under control of governor, including state universities
January 4, 2011
https://www.lawlogix.com/wp-content/themes/LawLogix-Nova/resources/PDF/STATES/FL_11_02.pdf

8.  University of Maryland - MD - No E-Verify law

9.  University of Minnesota (public university) - MN - MANDATORY for public agency (January 7, 2008 - April 4, 2011)
July 20, 2011: "Service contracts over $50,000 require certification from the vendor (and subcontractors), as of the date of services, that it has enrolled and is implementing... E-Verify for all new employees performing work on behalf of the State."
https://www.lawlogix.com/wp-content/themes/LawLogix-Nova/resources/PDF/STATES/MN_08_01.pdf
https://www.lawlogix.com/wp-content/themes/LawLogix-Nova/resources/PDF/STATES/MN_16C.075.pdf

10.  University of Missouri (public university) - MO -  MANDATORY for public employer
January 1, 2009
https://www.lawlogix.com/e-verify-map/missouri/

11.  University of North Carolina - NC - MANDATORY
March 1, 2007
"All state agencies, offices, and universities must use E-Verify for new employees hired on or after Jan. 1, 2007, except for employees of local education agencies hired on or after Mar. 1, 2007."
https://www.lawlogix.com/e-verify-map/north-carolina/

12.  Vanderbilt University - TN -  MANDATORY
January 1, 2012
https://www.lawlogix.com/e-verify-map/tennessee/

It is possible that of Beemer's 12 examples of universities using E-Verify campus-wide, a lower majority and not 10/12 are doing so because of state E-Verify policies. It is also possible that 10/12 are using E-Verify because of state policies.