The company recommends using an external hard disk that is wall-mounted (using AC power). The Wii U’s maximum storage capacity is 2 TB, however you should never require that much. Devices with larger storage capacity will function, however the Wii U is limited to 2TB. Additionally, Nintendo discourages the use of external SSDs.
The website of Nintendo contains a list of external drives that have been confirmed to be compatible with the Wii U. It should be fine to use any of them, however notice that the majority of them are outdated models. We elaborate further below.