bus from DC to union city,NJ

I doubt if you'll find a bus directly to Union City from DC. Your best would be to take a bus to Philadelphia or NYC depending on whichever is closer to this place, and taking SEPTA and/or NJT or Amtrak from DC to Trenton/Metro Park/Newark/NYC followed by bus/get picked up. The buses usually have a tiny restroom in them, but I cannot tell how the cleanliness will be. It would depend largely on your co-travelers.

If cost isn't an issue, I would strongly recommend (since you'll be with your very young daughter) to take Amtrak to the nearest station and try to get other modes of transport from there. Your daughter may find waiting in bus stops very difficult in the cold coming from CA, I'd imagine.

HTH.