Here is a very simple script (actually, you can really do the whole thing with just the one line) to remove files from a given directory that are older than a given number of hours. Adjusting the two variables to match your system should work. The third variable allows you to either delete everything, or delete only files matching a specific pattern (I.E. "logfile-*")
Once you have things sent the way you want, you could put the "find ..." line in your boot scripts with the minutes, directory, and file pattern hard-coded in.
Code:
#!/bin/sh
# Very simple script to handle old log files.
# Lifetime of log files
HOURS="2"
# Directory to clean
TARGET="/tmp"
# Filename to match
NAME="*"
find $TARGET -name "$NAME" -mmin +$(($HOURS * 60)) -type f -maxdepth 1 -delete
# EOF