Does eclipseProject task automatically add checkstyleNature to project?

There’s no built-in support for checkstyle, you have to configure the custom natures/build commands in your build script like you suggested:

 eclipse {
  project {
    natures 'net.sf.eclipsecs.core.CheckstyleBuilder'
    buildCommand 'net.sf.eclipsecs.core.CheckstyleNature'
  }
 }