I would change:

# Encrypt file.
my $gpg_command = `gpg --trust-model always -er 12345678 my_file.txt`;
print defined($gpg_command) ? $gpg_command : "Command failed\n";

# Capture STDOUT and save it to a file.
open(READ, "$gpg_command 2>&1 |") or die "Failure on open $!\n";

 open( my $out, ">", "encrypted.pgp" ) or die "Could not open file for
+ encrypted data - $!";
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to:

# Build encryption command
my $gpg_command = "gpg --trust-model always -er 12345678 my_file.txt";

# Execute command and capture STDOUT.
my $gpg_results = `$gpg_command`;
print defined($gpg_results) ? $gpg_results : "Command failed\n";

# Save captured STDOUT to a file.
open( my $out, ">", "encrypted.pgp" ) or die "Could not open file for 
+encrypted data - $!";
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Then:

• Correct any errant assumptions about what happens when the command fails;
• Adjust what you're writing to the output file to reflect what you actually want there.