I am trying to install a washer in my basement for a in-law apartment. I am installing this in a room in the basement directly beneath my current washer and dryer. I have read about the enclosed sump pumps. Can I use something like this to connect to the drain pipe coming off my current washer? Would I also need to put in check valves?

In you case this would be considered an ejector pit rather than a sump pump. Yes you can do this and yes you need a check valve. The pit will need to be vented and you drain coming out of the pit will need to be trapped if its going to the sewer line.