If the file is empty, the first attempt to read from it returns false (undef) and you never execute the body of the while loop. Your call of &stopit is inside that body so it never gets called. You can check for an empty file with eof before executing the while.

open(FILE, "links.dat") || die "Can't open file!";
flock(FILE, 2) || die "Can't lock file!";

stopit if eof FILE;

while ($line = <FILE>) {
....
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