Can I install both Dropbox and Box Desktop clients at the same time on a windows 10 computer? Will there be any issues with overlay icons or anything else?
As long as you keep the folder from one service out of the folder from the other server, you should be fine.
Overlay icons will most definitely be an issue, at least on Windows. Microsoft only allows a certain number of them to be registered, and Dropbox uses many of those slots. Whichever application is installed last will likely overwrite some or all of the overlay entries from the previous application, and you'll probably see them overwritten as each application is upgraded though normal use. There's nothing Dropbox can do about that as it's a Microsoft limitation.