Yes this is doable.
You will have to set up the share links on the relevant cloud services. So for dropbox users send them the dropbox link and for BOX send them their relevant links.
The syncing on synology is taken care of in the background and the synology box will be up to date with whatever changes they make and save to the respective cloud services.
I would recommend making sure the sync timing for checking changes is as close as you need to being real time so that you see what has been changed when. Do not apply encryption on the synology box otherwise no users will be able to access the files.
Apologies I misunderstood the request, but have left the top part for consistency, along with the above there are several other setups you need to follow.
Option 1 - One synology box.
Set up as above and create 2 separate folders on your synology box. One for BOX.com and one for Dropbox. Have these syncing bi-directionally.
Then use a folder copying app from one to another. I would recommend perhaps HyperBackup to backup one folder to the other every hour. If you need something real time perhaps look at option 2 or option 3.
Option 2 - Buy another synology box.
So with this you connect each synology box to the cloud service needed.
You can then use cloudsync between the boxes to make sure the folders are up to date with each other and that will work out changes for you.
Option 3 - Use a computer - Not tested but should work in theory.
Set up cloudsync on your computer Mac or PC and select a folder.
Download dropbox and select the same folder as cloudsync.
Then those should be synced with one another.
Then on synology nas sync BOX to that folder as above.
The downside to this is that your PC would need to be one but would have close to real time as possible.
Option 4 - Send a feature request to synology to remove the restriction of same folder to 2 different cloud based solutions.
Hope that helps.
Any other questions let me know