find: how to not have errors (stderr) when no files exists

Try this

#redirect stderr to null
exec 2> /dev/null
FILECOUNT = `find *.log -mtime +1|wc -l`

#redirection of stderr to file
exec 2> /tmp/errors.txt

# gzip old log files    
if [ $FILECOUNT != '0' ]; then
   find *.log -mtime +1|xargs gzip -f
fi

# test if error log file is empty
if [ -s /tmp/errors.txt ]; then
    .....
fi`

If you are using a GNU version of xargs you can use -r:

   --no-run-if-empty
   -r     If the standard input does not contain any nonblanks, do not run
          the command.  Normally, the command is run once even if there is
          no input.  This option is a GNU extension.

Code:

# redirection of stderr to file
exec 2> /tmp/errors.txt

# gzip old log files
find *.log -mtime +1|xargs -r gzip -f

# test if error log file is empty
if [ -s /tmp/errors.txt ]; then
    .....
fi