Open Database mysql.gz
The answer to another person's question about gz.cpgz file-types suggests that it may have been compressed twice. Other searches show people having problems opening them but this is down to software not expecting files to be zipped up twice in their cases.

You don't say what operating system you are on but, from what you've posted, I'll take a guess that you have access to the file command to help identify what type file you have.

file database-name.mysql.gz will show that you have a file that can be unzipped using gunzip as you've already done.

Now run file database-name.mysql.gz.cpgz to see what this file's type is. If it comes back as gzip file again then you can gunzip it again. 

A MySQL backup is just a text file, so if it reports it as so then you're safe to check it out further. Since it's over 300MB I wouldn't recommend opening it in your text editor. Instead use the head command to check out the top few lines:
head &lt;yourfile.sql&gt;

If you're Windows user 7zip will be able to tell you if the gz.cpgz is an archive or not.