If you are sure the CPU is idling at 60C, there is either something wrong with it, or the cooler, or its mounting. That high of temp could very well explain the shutdowns. I'm sure the load temp must have been near throttle if not beyond.
If you are using the stock cooler for the i7, check the fan RPM, see if it is spinning at or near the i5 cooler's RPM.
Also check the 4 posts that push thru the MB. Intel still has a rather poor design for mounting coolers. Be sure the posts' feet are flat on the MB and that the black pushpins are properly spreading the white fingers behind the MB.