Concerning mechanical Stress:
As Uwe has pointed out in the comments, there is no large mechanical stress on the probe at any point in the mission after launch.
The thrusters cause stress, but not more than a theoretical combined maximum of 13.3N distributed among 16 thrusters. The load-bearing structures are designed to handle this.
During the entirety of a gravity assist, the spacecraft is in free fall, so it will not experience any acceleration. There will only be a tiny tidal force due to one end of the spacecraft being slightly closer to the central body than the opposite end.
As for thermal stress, heat soak is an issue, but the manufacturer of the thruster makes sure that the thrusters are adequately cooled by radiation. Other components can see some heat radiation from the thrusters' plume, but it is important to note that these things are tiny, and do not produce a lot of heat to begin with.