You shouldn't feel guilty at all. They should feel guilty for making you need to take that day off.
Your workplace should be a refuge, where you charge your batteries and build sanity. Not a constant drain on those resources.
That's a refreshing perspective, I haven't thought about it like this. Indeed, when you're solving tough problems, you'd better concentrate at the private office (which is nowhere to be found, so work from home it is), and the workplace is the place where you go to discuss your difficulties with colleagues to broaden your perspective, and ask senior ones for practical advice. Makes perfect sense.