If these mice have all male offspring, why won't they be out-bred by the mice that have females too? Why would a non-advantageous mouse gene be passed down and take over? Wouldn't natural selection kill off the genetically modified mice?
The answer to your first question answers your entire set of questions.
The basic driving force of evolution is reproduction. Fitter animals should produce more and fitter offspring, whereas less fit animals will either produce no offspring, or will produce less-fit offspring. That is one of the most basic premises of evolution.
The issue here then is "fitness", and whether or not the modified mice will have sufficient fitness to a) reproduce, and b) introduce their genes into the next generation.
The modification only changes the outcome of birth -- all mice fathered by the modified mice will exhibit the same modification, and will be born male. It doesn't impact their ability to reproduce, or their ability to fill their ecological niche. The mice will be at no reproductive disadvantage when compared to non-modified male mice, in that they will be just as likely to survive to reproduce, and will not have a shortened lifespan that causes them to reproduce any less than a non-modified male. Thus they won't be out-bred; a female mouse isn't going to have any way to distinguish (at an evolutionary level) between a modified and non-modified male. Now if the modified males also glowed bright green and failed to attract female mates, then you'd have a situation where the modified males would be at a disadvantage, however, that isn't the case here.
Not only will there will be no evolutionary disadvantage to the modified males in terms of reproduction, over time they'll actually have the advantage. Assuming litter sizes average out the same, ALL the offspring of modified males will also be modified males. Let's call that average M. The offspring of unmodified males will be mixed male and female; the average number of unmodified male mice offspring will be M/2 (as half will be male, half female). The modified mouse will have double the male offspring of the unmodified mouse. The population of modified male mice will increase linearly, whereas the population of unmodified mice will (at least initially) be relatively stable.
Over a longer time period, female mice will be more and more likely to mate with modified male mice, as they will be more available. I essence, this gene modification hacks evolution by making the modified mice MORE fit than the unmodified mice, in that their offspring will be more competitive in terms of mating with females, due to sheer numbers. As females die and are replaced with fewer and fewer females, and as the modified male population continues to soar, you're eventually going to get to a point where the only available males in a community to mate with the few remaining females is going to be modified males, who will only produce male offspring. Those last remaining females will eventually die off, and with no new females within a given local population, no further reproduction can occur, at which point the population of remaining males eventually dies off.
(I do note a "local population", as this only works within populations that reproduce together. Geographic or other divisions in reproductive populations may cause certain islands of mice to continue unaffected if there isn't a critical mass of modified males. So if the country mice and city mice don't reproduce together, one or the other may be unaffected if the modified mice aren't artificially introduced).
All of which would make for an interesting computer simulation. I may have to get on that this weekend.
All that said, it will be interesting to see what behavioural changes may be introduced in newer generations as the number of males begins to strongly outnumber the females, and opportunities for the males to reproduce decreases. Will male mice become more territorial? Mouse combat to the death for access to females? If they go ahead with this plan, I hope there is funding somewhere to study the behavioural changes as time progresses.
 - In popular culture, the idea of "Darwinian fitness" is often confused with strength, physical fitness, or lifespan; the concept is really about the ability of an individual to pass on its genes to as many individuals as possible by fitting best into a suitable ecological niche.