Republicans Should Support Third Parties...
...but not vote for them.
The political structure in the United States necessitates a two-party system. With current structures, a third party will either fail or become one of the two parties. However, should the political structure be changed to allow for a third party, the Republican Party would be the primary beneficiary.
The Democrats are a coalition masquerading as a party. Should multiple parties be viable, the uneasy bedfellows in the Democratic Party would split into separate parties, e.g., Greens, Labor, Socialist, a few ethnically-based parties, etc. They would be natural allies, but they would lose the economies of scale as each separate party would have to develop its own structures.
The Republican Party, by contrast, is far more homogeneous. It would probably lose a rump to a revitalized Libertarian Party, and perhaps a new hard right Conservative party would form, but it would easily swallow up the middle of the political spectrum. It could then wheel and deal with the splintered parties around it and dominate American politics for decades to come.
Divide and Conquer.