Issue with loading/using external javascript static resource in Lightning component when Locker service is Enabled

You just forgotten to declare the variable zipFile, change the line

zipFile = new JSZip();

to this line:

var zipFile = new JSZip();

also I will recommend you to use the Salesforce Lightning CLI to check this kind of issues, its a heroku command line tool that check for this errors.

Follow this steps to install Salesforce Lightning CLI: https://developer.salesforce.com/docs/atlas.en-us.lightning.meta/lightning/cli_intro.htm

UPDATE: In my personal org I tested the code you provided with a few modifications in an aura application and it worked fine. Here is the code:

App:

<aura:application >
    <ltng:require scripts="/resource/jszip/jszip.min.js" afterScriptsLoaded="{!c.onPress}"/>
</aura:application>

Js controller:

  ({
        onPress : function(component) {
            var packageXml = '<?xml version="1.0" encoding="UTF-8"?>' + 
                '<Package xmlns="http://soap.sforce.com/2006/04/metadata">' +
                '<version>35.0</version>' +
                '</Package>' ;

            var zipFile = new JSZip();
            zipFile.file('package.xml',packageXml,{base64: 'true'});
            var data = zipFile.generateAsync();
            alert(data);
        }
    })

I downloaded the last version (3.0.0) of jszip from their site and had the issue that the method generate() is deprecated, so I changed it to generateAsync() as is suggested in the upgrade guide.