Just move your if outside the loop:

foreach $logrecord (@errorcheck)
{
    open (LOGFILE, "$logrecord");
    while (<LOGFILE>)
        {
        my $logline = $_;
        if ($logline =~ /$error/)
            {
            $errorcount++;
            }
        }
}
system ('page Mike') if $errorcount;
print "$errorcount\n";
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Or did I misunderstand your question?

By the way, use != instead of ne when comparing numbers. Otherwise negative quantities will surprise you.