You need to save the args too

my @query_queue;

sub do_query {
   my ($sth, @args) = @_;
   push @query_queue, [ $sth, @args ];
   flush_queries();
}

sub flush_queries {
   while (@query_queue) {
      my ($sth, @args) = @{ $query_queue[0] };
      ...[ attempt query ]...
      if (...[ successful ]...) {
         shift @query_queue;
      } else {
         return;
      }
   }

   return 1;
}

do_query(...);
do_query(...);
do_query(...);
flush_queries()
   or die;
[download]

Depending on the reconnect code, it might be more advantageous to pass the dbh and the stmt instead of the sth, but you get the idea.