I think someone can be both
However, If they don't want to be part of the republican party ... then leave
I'm sure their new found buddy and hero, Pat Buchanan would love to have them
My house has a leaky roof. I'd rather fix it than abandon it.
The same can be said for my party.
Then the party will be empty. A party is just a collection of people, nothing more. The coalition adopts some common ideals which in the case of Republican party are conservative.
Parties change, ideologies stay pretty stable. The Republican party of 2004 is far different than the Republican party of 1968.
If the Republican party wants to adopt liberal ideology as Bush most certainly has done in fiscal matters, then the Republicans can't expect to get many voters with a conservative mindset to vote for them.