1 Week in France

I am basing this on what you are asking not what would seem realistic to me.
I am assuming you already have a round-trip ticket to Paris from somewhere, so I would get the order together first something like this :

June 4, 5, Paris
June 6th take the earliest flight to Nice (stay in Nice).

It takes too much energy and time to drive or train from Paris to Riviera ; the norm is that you will be sleep deprived jet lagged the first days not unless you are traveling to France from Europe.

June 7th Nice include some day trips I would go at least to Antibes and Monte Carlo using the train
June 8th Nice day trip to Eze by bus
June 9th fly back to Paris
June 9, 10 Paris

Then find and book the flights roundtrip from Nice.
Then book the hotels.
Then research what you want to see with your time,
Then ask questions whatever is unclear

Have you purchased airfare yet? If you must see Paris AND the South, fly multi-city into Paris and out of Marseille or Nice (or vice versa) and save the back and forth travel. Honestly with one week I'd plant myself in Paris and have one or two day trips within an hour or two of town-there are a few great books on day trips from Paris-and see the South on another trip. Or, do the South and catch Paris later. Travel time = vacation time.

Start by being honest with yourself. You do not have "1 week in France". You have 5 days. 5 full days, on the ground, that are usable. You can't count your arrival or departure days as usable for much of anything (other than trying to stay awake after arrival, and getting ready to catch your flight home).

Where are you coming from, and to where are you going after France? (Makes a difference: if you're just coming from London, that's one thing; if you're coming from Australia, that's quite another). Have you ever been to Europe (or to a foreign country) before?

I think the #1 mistake that first-time travelers make is trying to "see as much as possible", so be careful. Unless you've been to Paris before and have seen enough of it, or you expect that you can and will come back any time you want (lucky you), if this is your first trip, I think that you should stick to Paris (giving that several days) and pick one, maybe two (at most) other places that are relatively close by.

If you have your heart set on "somewhere in the French Riviera" then you should go spend your 5 days there - I don't think you have time for other places around France.

The farther you travel and the more different areas you go to, the less you will see unless you count stuff observed from the window of a train, bus or airplane. That travel time is a dead loss. The way to see the most is definitely not to try to see Paris and the Riviera in five full days plus some additional hours on your arrival day (if you can stay awake) and your departure day.

It sounds like you already have your plane tickets. If that is the case, I would save the Riviera for your next trip and spend your time in Paris, choosing one or two day trips that align with your interests.

I have been to Europe before and it's an overnight flight from the US to Paris so when we land we will be fine to explore that day.... I have been reading and see there are overnight trains that go from Paris to Nice so I think it would be possible to spend a day or two there and then do another overnight train back. Thoughts?

Sadly the Paris-Nice overnight sleeper train was discontinued as from 10 December 2017.
It so much easier and cheaper to fly nowadays

it's an overnight flight from the US to Paris so when we land we will be fine to explore that day

You are lucky if you can sleep that well on the plane! I'm a mess on arrival day and would not try to take in anything substantial because I would forget what I saw.

Overnight trains, if they exist, are not the romantic and relaxing treat one might think. Fly to Nice or alter your plans. A week in Paris is delightful and such fun without the stress and time spent changing locations.

I have been to Europe before and it's an overnight flight from the US to Paris so when we land we will be fine to explore that day.

You could be the exception, but for the majority of people flying overnight from North America, your arrival day will probably be at least a bit of a struggle to stay awake until after dark. Realistically, even before they get on a plane, many people start a trip with at least a bit of a sleep-deficit (maybe a large one), due to the inevitable pile-up of last minute work and pre-trip tasks. Sleeping on flights these days is a real challenge for many (its much better in business class, but if you're crammed in coach, and especially if you have a connecting city or two which makes for shorter individual flights but a longer itinerary), add in pre-trip stress that tends to build up, excitement for the trip ahead, and it's just not a great environment for getting rest. Some people (few, IME) can get good sleep under these conditions, but be realistic. Everyone can tough-it-out for a few hours, but most will start to fade soon. Most people set a goal of simply staying awake until after an (early) dinner on their arrival day, and I think that's an appropriate goal. Stay outside in the sunshine and fresh air, and keep moving. Avoid time indoors, in quiet places (eg museums). Maybe you will enjoy wandering Paris for several hours, but keep your expectations low for what you can accomplish that day.

Previous advice stands. You have five days there, not a week. If you keep telling yourself you have "a week" you will likely try to do too much/go too far. With five days, I'd stick to Paris and a day trip or two. But it's your trip. Good luck.

Yes, I've attempted a trip like this and fell asleep at a concert that I really wanted to see on the 2nd day.