Deny is the opposite of accept.
Living in denial = "Living in a mental state where they did not accept".
However, it tends to be a bit stronger than simply "they did not accept", and normally means something like "they refused to accept", with the implication that there was plenty of evidence that meant they should have accepted.
The Milton one is more complex. Milton wrote it in "on his blindness", and the original meaning was that while many did great deeds in the service of God, the blind man who was not able to perform such deeds also had a place in God's service, and was equally valued despite his disability.
During the Second World War, however, the line was adopted as the motto of the British Royal Observatory Corps, who (in the days before radar) kept watch for enemy planes on bombing missions, and raised the alarm when they saw them.
Perhaps because of this, the phrase is now increasingly used to describe the value of those who perform non-military activities in a war.