Ok, that's either a rendering issue (the text isn't being rendered correctly) or it's a version issue and the UI calls aren't being handled correctly.
There are two things I'd like you to try:
1. Try to click on the top text field and carefully type out your email address. Then click on the bottom field and carefully type out your password. Then click on the button below both fields. You won't see a cursor or anything like that, but if it works and you can log in then we'll know it's a rendering problem that's somehow unique to your setup.
2. After trying the above, see if you can uninstall the unity plugin and re-download it again by visiting the play page. I'm running v 3.1.0f4 and it's not giving hassles. I'm wondering what the normal version downloaded is.