Set the correct server request path
Perhaps consider building an array of path segments through your logic and then setting page with it.

app.get('/*', function (req, res) {
  var page
  var base = req.params[0]
  var baseExtension = path.parse(base).ext
  // determine path segments
  var pathSegments = [__dirname]
  if (base !== '') {
    pathSegments.push(base)
  }
  if (baseExtension === '') {
    pathSegments.push('index.html')         
  }

  // set page
  // here we use spread operator
  page = path.parse(path.join(...pathSegments))
  // if not using ES6, you could alternatively use
  // page = path.parse(path.join.apply(pathSegments))

  // perhaps consider switch here instead of if-else if you are going
  // to have more than just html and css extension types
  if (page.ext === '.html') {
    /*... do html stuff ...*/
  }
  else if (page.ext === '.css') {
    /*... do css stuff ...*/
  }
  /*...*/
})


Be consistent on your usage or non-usage of semicolons to end lines of code.  It seems like you have chosen to use the no semi-colon style, so why is there one after var page;?