Most Republican's care about women's health issues, just like the health issues affecting men. Pneumonia is not just a woman's health issue; the disease impacts many elderly each year. Clinton is wrong to joke about her health condition, and she should be more forthcoming with the American people. Pneumonia can be serious for the elderly and is not something to joke about.
The only time that Republicans raise a concern over women's health issues is if there is a moral majority reaction. The major issues of birth control, abortion and low cost health coverage have been championed by the Democratic party. And it may take a woman in the White House to promote further issues for women.
No, Republicans as a whole do not care about women's health issues. There is more than likely a small faction of younger Republicans who do, but being that the Republican demographic is old Caucasian men who would rather see life go back to the days of Mad Men, women's health is not important to them. It's used during elections to gain votes, and then pro-choice laws and women's health research is eliminated all together.
Republicans are making a big deal now about women's health because they want Hillary Clinton to lose so they are using her health against her though. Historically though, Republicans can't even see past the small amount of money that Planned Parenthood spends on abortions, with the rest spent on women's health, especially poor women.