Light transport Captain reported the autopilot disengaged during arrival descent resulting in pitch down and overbanking. Captain recovered and continued the flight.

Narrative

We were descending on the JAIKE4 into TEB; but we were not cleared to descend VIA the Arrival. We were cleared to descend to FL200 and to maintain 280 kts at the transition. We were in FLC mode at the time and were descending to FL 200 as instructed. We were about 3 miles from SPNCR @ approx. FL 210 when we were instructed to cross JAIKE at 13;000 Ft. I started to set the VNAV to comply and saw that it was out of the range to couple this crossing restriction; we would have had to lose about 8000 Feet in approximately 8 nm. As soon as I realized the close proximity I disengaged FLC and engaged VS and increased the descent rate.I notified ATC that we were unable to make this crossing restriction and they told us to turn to a heading of 090. I disengaged NAV and engaged HDG (Heading) and commanded 090 heading. While I was consulting my arrival chart; the aircraft continued its turn beyond 090 heading by approx. 20-25 degrees and the nose had pitched further down; this happened over a matter of seconds. The Auto-Pilot had disengaged without any Aural Warning. In the seconds that it took to realize the Auto-Pilot disengagement; I rolled wings level and recovered the aircraft and continued the flight; no excessive 'G' Loading was applied; the nose had dropped to approx. 12 - 13 degrees down and bank had increased to approx. 45 degrees. We never received a bank warning and we were too high above the ground to receive a 'Sink Rate'. The whole event was over in less than 30 seconds. The passengers didn't comment and when I asked them; they thought we just had a little turbulence.A couple of lessons learned. First; when descending on an arrival; whether descending VIA or to a cleared altitude; descend in a manner to make all crossing restrictions as if you were descending VIA. Second; it only takes a couple of seconds for an upset to occur; vigilance of the Auto-Pilot is paramount.