Hi,
I also had trouble with the Yubikey Neo but I got one-touch 2FA working by doing this:
1) Register key for 2FA in GitHub (I don't have Facebook but I assume it's the same)
2) Install Google Authenticator on your phone - it wouldn't work without this
3) Open GitHub in Chrome (mobile), enter username and password
4) When prompted, wave the Neo over the back of the phone 20 or 30 times
I had trouble getting the phone to detect the Neo, probably due to my massive phone case - but it does work with just a tap. The first time I got sent to the same yubico page as you did, but I closed this without installing any apps and NFC is working fine.