General aviation pilot reported a near miss during cruise with an aircraft conducting a missed approach.

Narrative

I was flying with my safety pilot at 7500 over ZZZ. Our plan was to fly east of the airport at 7500 and avoid traffic. As we were flying over the airport; we were advised that someone was going missed at ZZZ. After reviewing both the VOR and GPS missed approaches it appears the published approach has the aircraft going to the right (west) of the airport to intercept points on the approach path. From what I can tell; reviewing my flight log; it appears they would have had to turn right and maintained 7500; but they should have maintained 7300 according to both published missed approach procedures.We flew at 7500 and were clear of the traffic. I then was told by my safety pilot to turn left heading 270 quickly and began to do so until I noticed the aircraft was in that direction. I snapped that he was turning me into traffic; just then ATC told us to immediately climb to 8000 ft to avoid the traffic and while going 100 kts indicated I pulled back pretty hard and climbed to 7800 while reducing speed to 70 kts indicated then began a left turn towards 090 to avoid the traffic which was off our 270.ATC then advised we can continue own navigation. The aircraft got very close to us; and I'm not sure just how close but I feel that multiple factors had led to this near miss. The safety pilot could have had better eyes on the traffic below and realized this C172S was climbing towards us; the published missed approaches call for a right-hand turn; not a left-hand turn; so turning the wrong way put us directly into each other's' path and ATC who was in contact with us and the pilot on missed could have coordinated that better; even if we were simply a VFR pilot on flight following; we depend on ATC to advise of traffic and we will take all orders from ATC. Luckily; ATC told us to climb immediately; the other pilot was at 7500 headed directly for us and I believe the extra 300 ft saved us from a mid-air collision.