{{#paper-select
    placeholder="Select a thing..."
    required=false
    allowClear=true
    ...
}}

This option is passed along to the ember-power-select component that ember-paper uses under the hood, so the resulting clear button can be styled accordingly. For example,

.ember-power-select-clear-btn {
  position: absolute;
  right: 25px;
}

Thanks to @miguelcobain for this tip on the community Slack channel.