When transposing, if its transpose down a Major 6th from C. Do you take the key of Eb and say that a major 6th up is C as that would be a major 6th down. But cause your starting note is C if you took the key of C major it would be E natural?
Yes, Eb is correct. I'll offer another way to think about it: change the interval to half steps. Since a major 6th is 9 half steps no matter what key you're in, transposing down a major 6th means transposing down 9 half steps. It doesn't matter that the Eb is a minor key in C ... you're not in C anymore. C has gone from the tonic to the major 6th of Eb. Same pitch, different context.
My rule of thumb in general, however, is not to go over 6 half steps. So with your assignment, I'd actually invert and think about it as composing up a minor 3rd, which is 3 half steps and the exact same thing. (NOTE: this would be an octave higher of course, so this rule of thumb only applies when octave doesn't matter, such as transposing chord progressions. If you're doing a melody, the octave matters very much.)