Even if using your personal laptop while hard-wire connected to an employer's/prospective employer's network or using a wi-fi connection at your employer/prospective employer, the employer/prospective employer may log/view all connections made through their network.

Private browsing keeps certain information off your hard drive, but the connections made through your employer's/prospective employer's network may be logged by the employer/prospective employer. That is how employees get caught viewing inappropriate content (porn and other content) at work.

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The solution to your problem is ANONYMOUS WEB BROWSING see this:

Anonymous Web Browsing

http://en.wikipedia.org/wiki/Anonymous_browsing

How to do Anonymous Browsing on Firefox ?

http://www.madrasgeek.com/2010/09/how-to-do-anonymous-browsing-on-firefox.html

Check and tell if its working.

you can also see more suggestions here:

http://www.google.com/search?q=anonymous+browsing+firefox&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a

My situation is that I am at home working on my network but logged into a prospective employer's website to fill out a job application. Can they violate my privacy and collection information, such as what sites I have browsed? If i do not install any programs from their website, would they have any ability to collect private information from my computer? Wouldn't collecting any data at all be a violation of privacy without agreeing to this?

They might be able to view your cookies and maybe some browsing history.

Even without installing "their software" they could do the above, although the legality is questionable - but I'm not a lawyer and this isn't a solicitation for selling you legal services.

That would be hard to prove even if they did that.

My advice is to clear your browser cache and cookies before going to that website to play it safe. Or use a web browser that you don't usually use, which might have little in the way of cache or cookies.

Another solution would be installing Portable Firefox to your hard drive. It will install completely separate from your regular version of Firefox and won't use any data from the regular version. Use that for your job app - no cookies, no browser history - no problems like you brought up.
http://portableapps.com/apps/internet/firefox_portable