filepicker.js

    filepicker.pickAndStore(
      {
        multiple: false,
        maxFiles: 1,
        services: ['COMPUTER', 'GOOGLE_DRIVE', 'DROPBOX']
      },{
        access:"private"
      },
      function(InkBlobs){
        var fileTotal = 0;
        _.each(InkBlobs, function (el) {
          fileTotal += el.size;
          Session.set('fileTotal', fileTotal);
          el.size = bytesToSize(el.size);
          var fullName = el.filename;
          el.mimetype = fullName;
          el.filename = longFileName(el.filename);
        });
        Session.set('files', InkBlobs);
      },
      function(FPError){
         if(FPError && FPError.code !== 101)
          showError(FPError.toString());
      }
    );

Once filepicker returns the array of Blobs, pass them to JSZIP to combine then pass the combined zip back to filepicker.pickAndStore()

Here are the filepicker docs:
https://developers.filepicker.io/docs/web/javascript_api/#pickMultiple

JSZip has a good example of how to bundle them up:
https://stuk.github.io/jszip/documentation/examples/downloader.html

@dylansimpson thanks for that. Sorry it took so long, had to focus on implementing other stuff.
So from my understanding, I have to modify this

      function(InkBlobs){
        var fileTotal = 0;
        _.each(InkBlobs, function (el) {
          fileTotal += el.size;
          Session.set('fileTotal', fileTotal);
          el.size = bytesToSize(el.size);
          var fullName = el.filename;
          el.mimetype = fullName;
          el.filename = longFileName(el.filename);
        });
      Session.set('files', InkBlobs);
    },

to

      function(InkBlobs){
        var fileTotal = 0;
        _.each(InkBlobs, function (el) {
          fileTotal += el.size;
          Session.set('fileTotal', fileTotal);
          el.size = bytesToSize(el.size);
          var fullName = el.filename;
          el.mimetype = fullName;
          el.filename = longFileName(el.filename);
        });
      // zipping it up
      var zip = new JSZip();
      zip.file(InkBlobs);
      Session.set('files', zip);
    },