Captain reported a NMAC with another aircraft and the need for evasive action after the opposing aircraft deviated from their assigned altitude.

Narrative

On Date conducting a flight from ZZZ to Billings; MT. Well inbound to Billings; ATC advised to stay at 7;000 ft.; fly heading 190; and traffic at 12 o'clock 5;500 ft. [We] had the traffic in sight (Aircraft Y; Piper Archer). When roughly 15 miles to the northeast of Billings; the traffic began to climb; but it did not look as though it was on a collision path. So [we] continued at 7;000 ft. and heading 190. The traffic continued to climb and began make a steep turn towards the [our] flight. ATC gave a traffic alert to Aircraft Y and told them to 'descend; descend; descend.' To avoid a mid-air collision Captain disconnected the autopilot; deviated from the IFR instructions; initiated a climb right turn. During one point the two aircraft were within 300 ft. of each other. Aircraft Y deviated from its assigned altitude. I believe we did take the correct precautions to prevent a mid-air collision. First Officer followed up with the Approach Controller via telephone. The controller said that Aircraft Y deviated from its assigned altitude and that we were not in the wrong. The Controller intends to follow up with the [FBO] to discuss the near-miss.