You need to change the format of your command line. Because the ">" is not inside quotes it will be interpreted by the shell, redirecting the output of the command to a file named value_2. This happens before your script gets control. You might try changing the command line to

-c:'field_1=value_1','field_2>value_2'
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Update: after looking at Getopts::Long again, it needs to be more like this, with repeated -c options:

-c 'field_1=value_1' -c 'field_2>value_2'
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This can be parsed with

use Getopt::Long;
our @opt_c;
GetOptions("c=s@");
print join ", ", @opt_c;
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but you will still need to break out the field names and values from each element of the array @opt_c with a regex, something like

/^(.*?)([<=>])(.*)/
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