Zwitterionic amino acids are routinely detected by HPLC and by MS. The only challenge is to find an appropriate stationary phase. The most common stationary phases for this type of separation would include octadecyl-derivatized silica (ODS, also known as C18) and a HILIC stationary phase. Many column vendors make columns explicitly marketed for separation of amino acids.
Sciex markets an LC-MS method using a C18 column.
Agilent markets an LC-MS method using a HILIC column.
A decently-cited paper from 2016 describes a 3-minute LC-MS method for amino acid separation using HILIC columns.
What needs to happen is that analytes need to reversibly equilibrate between the stationary and the mobile phase via a single "mode". If there are multiple modes by which analytes can interact with the column, then peak shapes will be highly distorted and/or flattened into noise. An example of this would be if the analyte can both interact hydrophobically with the C18 moiety of ODS and via Lewis acid / Lewis base interactions with bare silanols (Si~OH) groups on the surface of exposed, underivatized silica. These days, column manufacturers go through great lengths to insure that very few if any exposed silanols remain exposed on derivatized silica resin used for analytical LC columns. Zwitterions are really no different than other analytes in this regard.
At the MS, zwitterions will need to be transferred to the gas phase as charged ions for detection. But this isn't really a problem either, at least for e.g. electrospray mass spectrometry.