I would redirect my output in the file directly (as logging) like this:

#!/bin/csh

(
/bin/date

################################################################
# Recompile global QUICK screens, for Sydney
qdesign dict=/mis/sydmips/obj/phd.pdc << 'flag1'
use /mis/sydmips/obj/compilemips.qks nolist
exit
'flag1'
) > logfile 2>&1


or just for the HERE-document:

#!/bin/csh

/bin/date

################################################################
# Recompile global QUICK screens, for Sydney
qdesign dict=/mis/sydmips/obj/phd.pdc << 'flag1' > logfile 2>&1
use /mis/sydmips/obj/compilemips.qks nolist
exit
'flag1'