The NASA public affairs department may have exaggerated a bit with the "went wild" quote. The encounter was not even noticed until two years after it happened!
Evidence of the encounter was not noticed until 1998. Astronomers analysing old data found that Ulysses' instruments had detected a large drop in the number of protons passing, as well as a change in the direction and strength of the local magnetic field. This implied that the spacecraft had crossed the 'wake' of an object, most likely a comet; the object responsible was not immediately identified.
In 2000, two teams independently analyzed the same event. The magnetometer team realized that the changes in the direction of the magnetic field mentioned above agreed with the "draping" pattern expected in a comet's ion, or plasma tail. The magnetometer team looked for likely suspects. No known comets were located near the satellite, but looking further afield, they found that Hyakutake, $500 \cdot 10^6$ km (3.3 AU) away, had crossed Ulysses' orbital plane on 23 April 1996. The solar wind had a velocity at the time of about 750 km/s (470 mi/s), at which speed it would have taken eight days for the tail to be carried out to where the spacecraft was situated at 3.73 AU, approximately 45 degrees out of the ecliptic plane. The orientation of the ion tail inferred from the magnetic field measurements agreed with the source lying in Comet Hyakutake's orbital plane.
The other team, working on data from the spacecraft's ion composition spectrometer, discovered a sudden large spike in detected levels of ionised particles at the same time. The relative abundances of chemical elements detected indicated that the object responsible was definitely a comet.
Based on the Ulysses encounter, the comet's tail is known to have been at least 570 million km (360 million miles; 3.8 AU) long. This is almost twice as long as the previous longest-known cometary tail, that of the Great Comet of 1843, which was 2.2 AU long.
The primary source link "15" cited in the Wiki article uses the expression "went haywire" which NASA PAO may have translated to "went wild"