1. Why would someone's political affiliation be relevant to whether they'd be good at performing their job?
2. While political party membership is not a protected class under U.S. anti-discrimination law[1], it may be strongly correlated with protected characteristics like religion or age. So, hypothetically, if by discriminating against Republicans you're also systematically discriminating against Christians or people over 40 (due to some correlation between these groups), this may be a basis for legal action against you.
Even if you never get sued, if word gets out that you're delving into employees personal lives in such intrusive ways, your reputation might (justifiably) suffer.
[1] https://en.wikipedia.org/wiki/Protected_class