The simplest counter-example I know is the following: [Hartshorne](http://www.jstor.org/stable/2373984) showed that if $k$ has positive characteristic, $k[s^4, s^3t, st^3,t^4]$ (which will be $X_{red}$) is a set-theoretic complete intersection (which will be $X$). The former is well-known to be not CM (cheapest proof: $s^4,t^4$ form a s.o.p but not a regular sequence). 

 

There are more examples of projective curves which are set-theoretic c.i. (you can  find quite a few papers). Among them the ones which are not arithmetically CM give counter examples via taking the affine cone.