Dirty Rotten Scoundrels, The Bridges of Madison County, Hedwig and the Angry Inch, and The Most Happy Fella also have songs with a country influence.
Shows that take place in a Southern and/or rural setting that are arguable (but sound more like musical theatre)... Oklahoma, Giant, Annie Get Your Gun, Paint Your Wagon, Waitress
My favorite country-themed Broadway song is "I'm Leavin' Texas" from The Best Little Whorehouse Goes Public. Dee Hoty is a hoot on the whole album and I prefer the score to the sequel over the original as it gives itself over to pure camp from start to finish.
Marie: Don't be in such a hurry about that pretty little chippy in Frisco.
Tony: Eh, she's a no chip!
Violet, Bonnie and Clyde, The Golden Apple and Seven Brides for Seven Brothers might count, to varying degrees (I'm not certain of the boundaries of country music as a genre). And although I don't know the following ones well/at all, I suppose Destry Rides Again, Calamity Jane (film), and Pump Boys and Dinettes might be relevant.
This only half counts because it only played off- not on-Bway -- FLOYD COLLINS -- though hardcore bluegrass aficionados would say it is not country music.