Initial question:
According to the QDM the Patient Characteristic datatype has an attribute of "Start DateTime" and an attribute for "Stop DateTime", HOWEVER, the Patient Characteristic Expired datatype only has individual "Date" and "Time" attributes.

According to the measure logic guidance PDF, "Note that for “A during B” to be true, A must have specified start and end times. Some data elements in medical records, for example diagnoses of chronic conditions, can be open-ended. Such an event can neither contain another event temporally, or be contained. In other words if a chronic condition, represented without a stop date is event A, there is no event B such that ‘A During B’, or ‘B During A’ is true."
Since there is no "Start" and "End" time on the "Patient Characteristic Expired" datatype, you cannot check for it occuring "During" the encounter.

Response:
There is no restriction that "during" can not be applied to patient expired per QDM. It looks like the existing wording in the guidance document might be misleading.

1) The meaning of each attribute varies but generally are supposed to refer to the clinically relevant time such as the admission datetime , discharge datetime, incision datetime, removal datetime, facility location arrival datetime and facility location departure datetime. Others such as the more generic timing attributes (date, time, start datetime, and stop datetime) are a bit more vague on whether they are looking for a) the clinically relevant time b)indicating which clinically relevant time when the data element may have multiple, or c) capturing more of a date/timestamp for when the information was captured. These attributes generally refer to the clinically relevant time but this is not explicitly clear in the QDM documentation.
2) The QDM’s Timing Relationships don’t require that you use the QDM timing attributes. So you can use “during” without explicitly using attributes like start datetime and stop datetime.
The CMS 2014 eCQM Logic Guidance document should provide some additional guidance on how to handle these calculations. I hope this helps."