Do regulations require electronic checklists to be displayed on a cockpit screen or EFB?
No, nothing in the CFRs requires an electronic checklist.  In fact, almost the reverse is true - from experience I know that part 135 operators need to justify the use of EFBs, and have it approved on their operating certificate.  (i.e. FAA doesn't force them, they must demonstrate that the devices meet applicable standards and basically ask for permission...)
And I don't know of any GA aircraft that have electronic checklists built into their displays, although I cannot say that the capability doesn't exist.
Although many airline and GA pilots fly with EFBs, a great many do not.  It would be a tremendous burden to owners and pilots, and a regulatory nightmare to attempt to enforce the use of electronic checklists.
And there is absolutely no compelling reason to do so... The battery will never run out on a laminated paper checklist, so why would you want to force someone to use a less reliable method?