What's the gist of this Executive Order, and why might my cousin characterize it as "evil"?

Obviously, the original 1984 order (EO 12472) had no reference to DHS. But EO 13286 issued in 2003 includes amendments to the older EO in section 46, adding references to that department and removing references to FEMA, amongst other minor changes.

I don’t know the OP’s cousin but I don’t see any need to assume this is a racial issue. Political partisans have been coming up with conspiracy theories like this for decades (long before we had a black president) in which they claim a presidential administration of the other party is planning on seizing power.

Seizing power, as in the future? There’s a black Muslim Kenyan in the White House – that ship has sailed.

Who will promptly move on to the next right wing conspiracy e-mail.

Wait, which is the clause authorizing the FEMA death camps, again?

It’s the one between the two fnords.