We present an analysis of the HI 21cm absorption in a sample of 101 flux-selected radio AGN (S_1.4GHz_>50mJy) observed with the Westerbork Synthesis Radio Telescope (WSRT). We detect HI absorption in 32 objects (30% of the sample). In a previous paper, we performed a spectral stacking analysis on the radio sources, while here we characterize the absorption spectra of the individual detections using the recently presented busy function. The HI absorption spectra show a broad variety of widths, shapes, and kinematical properties. The full width half maximum (FWHM) of the busy function fits of the detected HI lines lies in the range 32km/s<FWHM<570km/s, whereas the full width at 20% of the peak absorption (FW20) lies in the range 63km/s<FW20<825km/s. The width and asymmetry of the profiles allows us to identify three groups: narrow lines (FWHM<100km/s), intermediate widths (100km/s<FWHM<200km/s), and broad profiles (FWHM>200km/s). We study the kinematical and radio source properties of each group, with the goal of identifying different morphological structures of HI. Narrow lines mostly lie at the systemic velocity and are likely produced by regularly rotating HI disks or gas clouds. More HI disks can be present among galaxies with lines of intermediate widths; however, the HI in these sources is more unsettled. We study the asymmetry parameter and blueshift/redshift distribution of the lines as a function of their width. We find a trend for which narrow profiles are also symmetric, while broad lines are the most asymmetric. Among the broadest lines, more lines appear blueshifted than redshifted, similarly to what was found by previous studies. Interestingly, symmetric broad lines are absent from the sample. We argue that if a profile is broad, it is also asymmetric and shifted relative to the systemic velocity because it is tracing unsettled HI gas. In particular, besides three of the broadest (up to FW20=825km/s) detections, which are associated with gas-rich mergers, we find three new cases of profiles with blueshifted broad wings (with FW20>500km/s) in high radio power AGN. These detections are good candidates for being HI outflows. Together with the known cases of outflows already included in the sample (3C 293 and 3C 305), the detection rate of HI outflows is 5% in the total radio AGN sample. Because of the effects of spin temperature and covering factor of the outflowing gas, this fraction could represent a lower limit. However, if the relatively low detection rate is confirmed by more detailed observations, it would suggest that, if outflows are a characteristic phenomenon of all radio AGN, they would have a short depletion timescale compared to the lifetime of the radio source. This would be consistent with results found for some of the outflows traced by molecular gas. Using stacking techniques, in our previous paper we showed that compact radio sources have higher {tau}, FWHM, and column density than extended sources. In addition, here we find that blueshifted and broad/asymmetric lines are more often present among compact sources. In good agreement with the results of stacking, this suggests that unsettled gas is responsible for the larger stacked FWHM detected in compact sources. Therefore in such sources the HI is more likely to be unsettled. This may arise as a result of jet-cloud interactions, as young radio sources clear their way through the rich ambient gaseous medium.