When I tap on a link in any app it shows me installed browsers on my HTC One X. The list includes Opera, Internet, Chrome and Firefox. Is there a way by which I can include options of Chrome (Incognito) or Firefox (Private Browsing) in that list so that I can open links in Incognito / Private Browsing mode?
Unfortunately I don't think there's any way for a user of an app to do that, I'm pretty sure it would require the developer to add that extra "intent" as an option to their app before it could appear in that app chooser box.
There are apps that specialize in this behaviour though, and should appear as options in the app chooser box for web links:
InBrowser is "a fully featured incognito/private browsing experience", which should appear in that list of browsers and work as an incognito browser.
Incognito Browser "automatically operates in incognito or private browsing mode"