This is a great question.
It's the sign of the times, Ashtonspen. Employers will use it and they will make decisions based on it. Whether you give them information or not. Profiles can be viewed and where there's a will, there's a way.
I'm a parent of two boys and at their school we had a parental social media workshop where they gave so many examples of students applying to college being rejected because of things they have said and posted on line--straight A students too. So, yes, even if they do or do not have access they are checking and browsing.
Another employer that outright asks for Facebook information is law enforcement agencies. They will most definitely view and make decisions, about your character, based on these things.
Therefore, to answer your question, it's not a matter of should they, but they just do, one way or the other. I understand the concern, but let's face it, we live in this digital, social age and we must be socially responsible for what we do on line. Digital footprints last a lifetime.
Like Ms. Dora stated, do not put anything that you do not want to be seen on line. Period. It's simple and sound advice in a not so common sense society that we live in.
Thank you for this question. You should definitely create a hub about it!