Cessna CE-680 Captain reported landing safely after inflight shutdown of the right engine following surging and high vibrations that resulted in inflight rollback and relight. Reporter stated during descent the left engine began exhibiting some of the same issues that led to the shutdown of the right engine.

Narrative

On Day 0 on Aircraft X we experienced what we thought was an engine vibration/surge/power fluctuation in the right engine during the descent. This led us to request priority handling; divert; and shut down the right engine. After shutting down the right engine we started to experience the same vibration/surge/power fluctuation in the left engine leading us to believe we would not make our diversion choice of ZZZ and that we may lose the left engine. This caused us to look for the nearest airport and land as soon as possible. We were closest to ZZZ1 and ended up entering a Left downwind and landing on XXL.Here are more details about the day and flight.This was our third flight of the day. We started out in ZZZ2 in the morning and picked this aircraft up from another crew. We flew from ZZZ2 to ZZZ3 and refueled for our next trip. The next leg was from ZZZ3 to ZZZ4 and we refueled and planned for our departure to ZZZ5. Originally scheduled to depart around XB:00. We received a brief and the flight was now leaving 1 hour earlier at XA:00. Once our 2 passengers had arrived we briefed them and departed at round XA:00 from Runway XXL out of ZZZ4. During the initial climbout we flew [through] a couple clouds around 3500 to 4000 feet and were in the clouds probably less than 15 seconds. The climb out was uneventful and we climbed up to our cruise altitude of FL430.During cruise at FL430 we experienced an Amber ENG CNTRL L-R CAS Message around XA:42. This CAS message led to an additional CAS message being displayed. This was the White AT FAIL CAS Message. The current conditions at the time of the CAS message were as follows. We had the Autopilot ON at FL430 in VMC conditions and the RAT indicated -39. We were around 50 miles from ZZZZZ in our flight between ZZZZZ1 and ZZZZZ. The fuel temperature was indicating 25 in the left tank and 24 in the right tank. After the ENG CNTRL FAULT L-R message illuminated; I transferred controls to the second in command (SIC) and started to complete the ENG CNTRL QRH. We completed step 1-5 and step 6 was land as soon as practical. I then called Maintenance while enroute and spoke with them. While speaking to Maintenance the IGN started flickering ON the display for both the Right and Left engine. I thought this was either a false display or a result of turning the engine Anti-Ice ON. The IGN illumination was light a light bulb flickering. Maintenance had transferred me to the chief pilot and while I was in the phone with the chief pilot the Right and Left IGN stopped illuminating. The chief pilot asked what our plan was and I told him that we were going to continue to our destination. We could stay out of the weather and avoid the weather directly in front of us. We had no other issues and the aircraft seemed to be working normal except for the loss of the Autothrottles.Then initially our descent was fine and there were no issues. We then received a descent and a vector to our left. Around 30;000 ft we experienced an engine surge/vibration in the right engine. This was probably an engine roll back with an engine relight. At that moment we received a change of frequency and I switched over and requested priority handling. We continued to get this engine anomaly and we were looking for a suitable airport. We decided on ZZZ. I attempted to load ZZZ in the box so my partner would have guidance. Then ATC gave us a heading and we ran the Inflight Engine Shutdown QRC. We made sure to verify that we were shutting down the engine with the anomaly. We completed the QRC and shortly afterwards we started to experience the same anomaly in the Left Engine. I advised ATC we would not be able to make ZZZ and we needed the closest airport. We were directed to ZZZ1 and landed on Runway XXR. The winds favored XXR and we had the latitude and speed to make the airport in the event the second engine failed.Since the only operating engine was also having an issue we were preparing to lose that engine. We advised Tower that we may lose the Left Engine as well. Knowing we would lose flaps; speed brakes; landing gear; nose wheel steering; and brakes the altitude and speed we had guaranteed we could make the airport safely. We were fast on the descent through 10;000 ft and the thought of losing the second engine gave us tunnel vision and we became focused on ensuring we made the runway. My fear was that if we put the gear down too early and we lost the last engine we would lose too much altitude to make the airport. Additionally; if we waited too long to get the gear down and the lost the engine we would have to do an emergency gear extension in a short amount of time. When the power was pulled back the engine would have this rollback/relight event and was very nerve racking. We were fast so we elected to put the gear down to help us slow down for landing. We completed the Engine Inop Approach QRC and moved on to the Engine Inop landing QRC. We had selected flaps full and the landing QRC caught it and we moved the flaps back to the Flaps 2 position. We slowed the aircraft as best we could and we landed safely on Runway XXR. We taxied the aircraft clear of the runway and shutdown in the ramp.This event was nerve racking and we did our best to work the situation together. It was helpful that we briefed what we would do in the event we had a CAS message during our crew brief. We knew who would fly and who would read the checklist. We did our best to comply with all FARs; company procedure; and policies. If we didn't comply with an FAR; company procedure of policy it was unintentional and a result of use dealing with the situation and the thought of losing our only operating engine.