There are only two ways to enter: from a desktop computer, or using an iPhone app. While their desktop version at first appears to be a collection of Flash games, it turns out they're actually implemented using something called the Unity Web Player, which is not available for Linux (even though they themselves use several of Linux's open-source libraries, and they're aggressively hyping Android [Linux-based!] compatibility for their next major release). In any case, I'm not too keen on downloading and installing yet another piece of mysterious crapware that phones home. I can't imagine corporate I.T. people, public libraries, Internet cafés, etc. allowing something like this anywhere near their computers.
Unlike previous United online sweepstakes, there appears to be no alternate means of entry if you're unwilling or unable to give an untrustworthy third party free reign over your personal information ... or anything else accessible using your credentials.
It doesn't seem worth setting up a completely sandboxed environment just to be able to play for a chance to possibly win a 10% discount certificate for a future flight.