How many African American Republicans are serving in Congress? None.
How many Hispanic/Latino Republicans are serving in Congress? Four, I think, and they're Cuban Americans and some have said that they don't really qualify as real Latinos. (Ileana Ros-Lehtinen, the Diaz-Balart brothers, and Mel Martinez).
How many Asian American Republicans are serving in Congress? One (Joseph Cao - and he'll be gone in 2010).
How many Native American Republicans are serving in Congress? None.
How many women are Republicans are in Congress? 21 (out of 91) in both the House (17/74) and Senate (4/17).
How many Jewish Republicans in Congress are there? One (Eric Cantor).
How many LGBT Republicans in Congress are there? None.
Yeah I wouldn't exactly call the Republicans the party of the minorities but more so the party of old angry white men. Republicans don't necessarily hate minorities (aside from homosexuals); they just, well, how do I put this diplomatically, don't tolerate differences?
Minorities only like the Democratic Party because the Democratic Party panders to them and they believe in affirmative action instead of hiring a person based on their skill. The Republicans are not very popular with minorities because the Republicans don't pander to their every whim.
The Republicans lost the minority vote on purpose in the 60s in order to win the white southern vote. You see the legacy of that even today. When Sarah Palin says "Barack Obama is not a man who sees America the way
you and I do" it's an appeal to mistrust of people who are a different color and have a different background or culture. When Republicans complain about lazy minorities taking welfare money away from hard working American heroes like Joe the Plumber, it's an appeal to those same sentiments. When Republicans say that small rural white towns are the "Real America," that turns off minorities, including Asians.
Asian immigrants value education highly, and as long as the Republicans keep pandering to Evangelicals (yes, Republicans pander too, just to different groups) with things like creationism and school prayer, they will have problems with the East and South Asian vote.
Your point on Affirmative Action is valid when talking about Blacks and Hispanics but not valid with Asians because if anything, affirmative action should help Republicans with the Asian vote, because Affirmative Action hurts Asians badly.