This feels like playing 20 Questions; every response provides a little more information that moves us closer to the answer...
It seems that your peptides are particularly basic. That leaves open the possibility that they have acquired some anions from the mobile phase to serve as counterions. In this case that would be acetate. You didn't specify if the peptide peak and the negative peak were observed eluting directly off the WCX column/cartridge or if you observed them in the subsequent HILIC run. If it's the HILIC run, then the peptide may have depleted some acetate from the mobile phase and then eluted more slowly than the portion of the mobile phase now lacking in some acetate ion. That would result in a negative peak sometime before your peptide peak, assuming you're monitoring absorbance at a wavelength that's absorbed by acetate (< 240 nm). You didn't tell us what wavelength you are monitoring.
Another possibility: Since your WCX mobile phase has no salt added, then when a sample of this mobile phase (in this case, the fraction containing the peptide that just eluted from the WCX column) reaches a HILIC column that's being eluted with 20-30 mM acetate, then the slug of liquid from the WCX column will have less acetate in it than the surrounding mobile phase and will have lower absorbance, at least if you're monitoring a wavelength that acetate absorbs. That accounts for your negative peak. Do you still see it in a blank run (i.e., with no peptide)?
Tailing of basic peptides in HILIC or lack thereof: Tailing of these in RP has usually been ascribed to interaction with silanols. Traditionally this has been handled by including an amine in the mobile phase to saturate the silanols and shield them from the basic groups in peptides. As for HILIC: I assure you that basic compounds can elute in badly-shaped peaks there as well. It depends on how much salt you have in the mobile phase. For moderately basic compounds, the 20-30 mM salt that you're using suffices. For compounds with a higher charge-to-mass ratio, such as arginine, you should use 40+ mM salt. For an extreme case such as aminoglycoside antibiotics (kanamycin, etc.), then you need over 100 mM salt in order to get symmetrical peaks in HILIC. For a demonstration of this effect, see Fig. 14 in my 2008 paper that introduced ERLIC. The problem involves a situation where molecules of the same compound have more than one possible counterion. The resulting ion pairs differ in polarity (or, more precisely, their degree of hydration) and migrate through the column at different speeds. Result: Skewed peaks or even multiple peaks for a single analyte. Using a high concentration of salt in the sample solvent and the mobile phase insures that all of the molecules of the same type will share the same counterions. Don't blithely desalt a peptide on a C-18 cartridge using TFA and then dry it down and shoot it onto a HILIC column when the mobile phase contains acetate or formate.