From the context, I believe that Burgess is using "Christnique" in two different ways.
In the first phrase:
Without us how would the Christniques get on?
"-nique" seems to be used as a Latinised form of the common suffix "-nik", which the dictionary tells us is:
a suffix of nouns that refer, usually derogatorily, to persons who
support or are concerned or associated with a particular political
cause or group, cultural attitude, or the like
which is used to form words such as beatnik and peacenik. So Burgess is coining the term "Christnik" as a slightly derogatory term for Christians.
In the second use, however:
Then they practise the Christnique of loving your enemies.
"Christnique" seems to be used here as a portmanteau between "Christian" and "technique", to refer to the Christian doctrine of loving your enemies, as a tool to convert non-Christians by good example.