As has been previously stated there is no critical engine on a jet a/c.
However if you are flying a crosswind take off and the downwind engine fails (let keep ourselves to twin jets but the same principle applies) during the ground roll and the take off is continued the effect of the engine failure will tend to counteract the requirement for into wind rudder due to the crosswind. However as the aircraft becomes airborne the rudder will have to be re-applied to keep going straight (in the air that is - never mind the ground track at the moment although this is also important).
I remember well a B707-436 which was written off during training at Prestwick in the mid 1970s. They were departing on RW 13 and the average wind was circa 220/15 (although the speed and direction were quite variable). After V1 the training captain closed the thrust lever on the No1 engine to simulate an engine failure. The trainee was slow to apply corrective rudder and as the a/c became airborne it started to yaw rapidly at which point the training captain took over control and (if my memory serves me correctly) restored the thrust on No 1 and retarded thrust on No 4. The main gear then contacted the runway again (lots of drag as much aileron in use) and folded sideways at which point a decision was made to abandon the take off. The a/c came to a halt a few yards from the fire station at PIK - all the crew got out but the a/c was destroyed (by fire I think).
One of the interesting factors in this accident which was pointed out in the investigation was that the VMCA wings level was 40 kt HIGHER than with a few degrees of bank towards the live engine. If the a/c had been banked a trifle towards the live engine they would have regained directional control and they might have got away with it!
Another factor was that the B707-436 was not fitted with a series yaw damper so all take offs and landing were made with the yaw damper disengaged which made the a/c somewhat of a challenge from a control point of view with engine out and/or crosswind.