Are there people who will refuse to use a website just because that site sends browser information to Google Analytics?
Could the +1 button be a red flag?
Should I expect that there would be similar thoughts around Facebook Connect?
This depends on your target group.
In Germany, for example, the privacy advisor of several states have claimed that the Facebook like-button violates German privacy laws. The threats by the privacy advisor of the state Schleswig-Holstein, to fine the governmental organizations and companies, which do not remove those things, got wide media attention.
Still most people will not care at all. But if you have a target group, that is very aware of privacy issues and Facebook repeated violation of the laws, this might have a negative impact. Although it is more likely that his group is using browser extensions to suppress the social plugins anyway.
Heise developed a two clicks for more privacy approach, which only loads the social buttons after the user clicked on a placeholder.
As far as Google Analytics is concerned, you can probably get similar information by analyzing the server log files. There is a number of tools available for that.
But it may be an issue, if Facebook is the only way to authenticate.
(Since you asked about the public opinion, all links go to press coverage. Unfortunately, I am primarily aware of the press coverage in my own country, so those links go to German sites. Google Translate might be helpful).
Frankly, I think it depends entirely on what your site is doing and who the users are that are using the site.
I won't object to Analytics (I use it on my sites) or Facebook integration for things that are social in nature. But, if my bank used Google Analytics (or Facebook connect, etc) within their banking portal I'd switch banks immediately.
If I could only log in via Facebook I wouldn't use the site. In fact, I couldn't because I don't have a Facebook account.
Your question is a bit too vague to really answer otherwise.
I would imagine that Google Analytics and +1 are fine.
I would imagine that FB Connect would not deter people if you provide an alternative way to log in. On the other hand, if the only way to use the site is to log in via Facebook, then some people might avoid using your site.
I have no data or measurements to support any of these claims.