I recently had to solve the same issue and had found a solution using JSZipUtils. 
The solution can be found here http://plnkr.co/edit/vWARo0dXbkgzmjyoRNi0?p=preview
I have two files that I would like to zip and download via the users browser and I call the function getBinaryContent(path, callback) on both files. The path here is the where the file is stored.
The two files are a .wav file and a .json file. Each of them should be treated differenly and hence you should use {base64:true,binary:true} for the .wav file and {binary:true} for the json file as an argument for the function file of JSZip.
The code can be found here as well
var file_confirmation=[false,false];
var file_name=["test1.wav","test.json"];
var urlList =["test.wav","test.json"];
var filenameSave ="myZip";
function zipFiles(id,urls)
{
    zip = new JSZip();
JSZipUtils.getBinaryContent(urls[0],function (err, data) 
{
    if(!err)
    {
        var dic={base64:true,binary:true}; //WAV File Encoding
        zip.file(file_name[0], data, dic);
        file_confirmation[0]=true;
        downloadZipIfAllReady(id);
    }
});
JSZipUtils.getBinaryContent(urls[1],function (err, data) 
    {
        if(!err)
        {
            var dic={binary:true}; //JSON File Encoding
            zip.file(file_name[1], data, dic);
            file_confirmation[1]=true;
            downloadZipIfAllReady(id);
        }
    });
}    
function downloadZipIfAllReady(id)
    {
        if(file_confirmation.every(function(element, index, array) {return element;}))
        {
             zip.generateAsync({type:"blob"})
                .then(function(content)
                {
                  var a = document.querySelector("#"+id);
                  a.download = filenameSave;
                  a.href = URL.createObjectURL(content);
                  a.click();
            });
    }
}
$(document).ready(function()
{
 zipFiles("a_id",urlList);
})