As I understood you want the filename to be user=newvalue and the string "user=newvalue" to be written to that file:
#!/usr/bin/perl
#
#
use strict;
use warnings;
my $teststring = '"user={hhhfj}jshdfjhdfjh\="';
my $own_value = "newvalue";
if ( $teststring =~ /\"user=.*/ ) {
$teststring =~ s /\"user=.*/\"user=$own_value\"/;
my $filename = $teststring;
print $teststring, "\n";
$filename =~ s/\"//g; #you don't want " in an filename
open my $outfile, ">", $filename;
print $outfile $teststring;
close $outfile;
}