The white balance card on the ColorChecker Passport is spectrally neutral. It is deliberately not an 18%-reflectance card (it reflects considerably more than 18% of the incident light) so it can't be used for an exposure card without knowing what "Zone" that tone lies on in your camera at the current ISO setting. It does, however, reflect equally across the spectrum, so it can be used for white balance (and you get finer distinctions in value with tones above middle level).
The black-to-white squares on the "classic" card (the one with the "+" in the centre and corner brackets that's used with the profile creation app) are also spectrally neutral, and picking one of the two lighter greys (not the white) should give you the correct white balance.
On the "creative" card, most of the squares are intended to give you a warm or cool white balance. The row with the people icons will give you a shifted white balance for pleasing skin tones; the row with the landscape icons will give you more magenta or green in the mix when you shift white balance, in addition to warming/cooling the Kelvin temperature (which only affects the blue/yellow balance). The centre square in the "landscape" row is a neutral WB, essentially the same as the WB card.
Note that white balance is only part of thee equation if you're aiming for colour accuracy. Creating and using a camera profile is also part of the deal. Not only are there camera-to-camera differences and lens-to-lens differences in colour rendition, there are also some baked-in manufacturer's "looks". In some cases, they're significantly less biased than, say, the difference between Kodak, Agfa and Fuji films of a similar contrast and saturation, but they're still there. F'rinstance, your camera may be deliberately ignoring the greenish tinge because, under most non-foliage circumstance, green is kind of icky (to use the technical term), so the camera's electronics (and the default camera profile) may be biased towards ignoring greens with little saturation for aesthetic reasons.
So, to sum up: shoot the ColorChecker Passport with the "classic" and "creative" cards visible, then continue shooting as normal unless/until you change something (camera body, lens, lighting). If you really want to set WB in-camera, shoot the WB card. (If you use a grey card, understand that it's designed for exposure, not necessarily for white balance. They're usually close enough for most purposes, but unless there's a time limit on the card, it won't be suitable for accurate colour.) When you go to post, first create a camera profile (which will involve creating a DNG from the raw image of the Passport). Apply that profile, and open the image of the Passport in your RAW converter. If you want an accurate, neutral white balance, select the centre square of the "landscape" row on the "creative" card. Correct the exposure as necessary. Then apply those adjustments (white balance, profile and exposre) to all of the images taken in that series. Rinse and repeat with other series.