I am confused by something in the edgeR user guide in section 3.3.1.  The worked example indicates that each sample is obtained from independent subjects but the Sample name in the sample column seems to indicate that the Drug and Placebo samples are paired (i.e. Sample1, Sample2, etc.). 

Is this example for a paired or an unpaired design?

Here is the Experimental Design file.

> targets
Sample Treat Time
Sample1 Placebo 0h
Sample2 Placebo 0h
Sample3 Placebo 1h
Sample4 Placebo 1h
Sample5 Placebo 2h
Sample6 Placebo 2h
Sample1 Drug 0h
Sample2 Drug 0h
Sample3 Drug 1h
Sample4 Drug 1h
Sample5 Drug 2h
Sample6 Drug 2h

If this was originally for an unpaired design and I wanted to convert this to a paired one where the pairing was based upon the labels Sample#, would the correct design be the following?

Group<-factor(paste(targets$Treat,targets$Time,sep="."))
Patient<-factor(paste(targets$Sample))
design<-model.matrix(~0+Group+Patient)

For Reference, the original code used to generate the design matrix in the user guide was as follows.

Group<-factor(paste(targets$Treat,targets$Time,sep="."))

design<-model.matrix(~0+Group)

The example is unpaired, just as the text says. I agree that the sample labels are confusing, but they are not part of the experimental design and should be ignored. (In our copy of the user's guide, we removed the sample labels from this example a while ago, but have not yet uploaded the edit to Bioconductor.)

No, it would not be correct to convert this example to a paired experiment in the way that you propose because differences between Times would be confounded with differences between Patients. If you have a paired experiment, please see Section 3.4.1. If you have a more complex design, you might also look at Section 3.5.