Have you kept your home warm since it's been cold outside? I'm in Ohio and I know Michigan has been cold lately that's why I'm baffled also.
Google Image Result for http://www.justbajan.com/health/articles/flyeggs/eggs1.jpg
This is off Google for images of what house fly eggs look like.
I'm sure you've already tried the litter box...
I once lived in NYC many years ago. The 1860 building I lived in had a swell roach problem. The way I solved the problem in my apartment was to leave all the windows open when it got real cold outside and spent the night at the neighbors elsewhere. When I came home all the roaches had gone...
Maybe you could do something along those lines but you would have to spend the night elsewhere if you kept all the windows open all night on a cold night. On a cold night but not so cold as to have adverse effects such as frozen pipes, but you could leave the cold water running to a fast trickle and be sure sink not clogged.
Take pets with you, of course.