The drain is located next to my washer in the basement but the overflow issue occurs when I use the slop sink on the other side or the kitchen sink directly above it, also when the water softner recycles at night(4:00am when I'm at work)! It dumps the recycle water in to the slop sink... what can I do to fix this or can I just try to plug or cap the floor drain?

You should call and have the city/township come out and check the sewage if it is really really bad.

I spent hundreds of dollars because the city told me it was my plumbing and couldn't possibly be theirs. They came out and looked in the open sewers and said they were right. Then a lot of neighbors complained and they finally came out and opened the manhole covers and the problem was a lot of garbage in the sewer system.

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what can I do to fix this or can I just try to plug or cap the floor drain?

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Sounds like a partial blockage downstream from the slop sink.
The solution would be to rent a sewer machine, (see image) and snake the kitchen sink roof vent. Put out enough cable to reach the base and 20 feet more. Good luck, Tom