The error message is stating that you do not have any space left, have you run "df -h" to see how much space is available on each partition so you can determine what needs to be cleaned out?

As far as I know you cannot tell squid to monitor the size of the log file but you can choose to use logrotate to cycle and compress the logs at regular intervals as instructed in http://www.linuxconfig.org/Logrotate, or you could also look at making a bash script that is scheduled through cron to check the size of the log file once and hour and if the file is past your chosen size transfer it to another server and create an empty file to replace it on the daemon server.