Is it common, uncommon or rare for a non-towered airport to have ILS (Instrument Landing System) runways? Can a non-towered airport have ILS?
Nontowered airports absolutely can have ILS approaches, and it's fairly common, though of course not as common as at towered fields.
Also, at many towered airports, the tower is not 24/7 and closes at night. If such an airport has an ILS, it's still available after the tower closes; ATC services for IFR approaches are provided by a nearby/higher facility like a TRACON or ARTCC, just like a nontowered field. Two examples of this in my neck of the woods are KCMI (Champaign, IL) and KBMI (Bloomington, IL).
I am not, however, aware of any Cat II or III ILS approaches at nontowered fields.
Protecting ILS Critical Areas is usually the responsibility of the tower, but even VFR aircraft are prohibited from entering the ILS Critical Area when the tower is closed:
(1) Localizer Critical Area. Except for aircraft that land, exit a runway, depart, or execute a missed approach, vehicles and aircraft are not authorized in or over the critical area when an arriving aircraft is inside the outer marker (OM) or the fix used in lieu of the OM. AIM 1-1-9(k)