Yes, the DOT rules do apply to CX on flights originating in the US and yes, the rules do apply to award tickets (referred to as "zero dollar"), and yes the likely compensation would be $1,300 (presuming that the cheapest ticket cost more than $325.

But, if the reason for the denial of boarding was crewmember illness resulting in the need to offload some number of passengers, CX may escape having to pay any compensation by asserting "operational or safety reasons" under 14 CFR 250.6.

The rules require that CX first seek volunteers and then offload passengers in some predetermined order. That could include fare paid, cabin class, status and so on, but it must be predetermined.

OP's friend should make a request to CX for compensation in the amount of US$1,300 (cash), wait a few days and then file a complaint with DOT. This can all be done online. Keep the complaint simple and to the point, e.g., "I was scheduled to fly on CX XXX on Date Y and was involuntarily denied boarding and rescheduled for 24 hours later. I requested and was denied compensation. CX did not request volunteers."

The rest is all fluff.