I kind of find your friends comment extremely rude. I mean yes, perhaps it would be distracting seeing someone at work. I for one would try and console them. Part of the reason I use to cry in the bathroom daily when I had internship while depressed. But her implying it's better for you to stay home isn't always the answer. Honestly I'm proud of you, you're still showing up and trying, however you feel.
And yes for some places there are mental health days. Here they are not a thing yet. You do have sick days, but depending on the type of job how you can use them. Some places would ask how sick are you etc and ask you to still come in if they don't have enough people--and saying you're depressed wouldn't cut it in as serious enough, sadly.
In some work places if you stayed home because of mental health they might take it as not taking your job seriously. Or overreacting. It's sad but true. The world is yet to catch up to mental health awareness, even more in workplace...