I don't think the two are related. The climate system typically
flashes the little dash lights over the recirculate and AC switches
when it detects a problem. It has it's own dignostic system.

The check engine light can turn on for hundreds of different reasons.
I don't see how they can say it is common until they read the diagnostic code.
Pepboys auto stores will read your diagnostic codes for free (for the
engine, not the climate control system). (I've never tried this; just heard
that Pepboys will do it.) The codes are mostly standard across all car
makes so they can look them up (orpost any codes read here for help).

If you enter "fan shuts off" in the search field (belwo, to the right)
you can find a few other posts with similar same fan issues.