First, yes, you are correct. "He" was improperly omitted.
Second, a bit of background: Different states have different rules about party affiliation and primary voting. In some states you must declare your affiliation with a particular party in order to vote in that party's primary. (For a counter-example, search for stories regarding Sen. Thad Cochran's recent primary battle in Mississippi.)
In general elections, users are allowed to vote regardless of any party affiliation. So Hunter may have meant to imply that he generally voted for Republicans, and was in that sense more ideologically aligned to the Republican party, but was not a "declared" Republican.
But nothing in the statement, "I used to vote Republican" precludes the possibility he was also an active, declared member of the party, either. And he could have been a registered Democrat and voted for Republicans in general elections.