Moving a large table to another filegroup
Yes your Foreign Key constraints (References) will work once you move the clustered index to a different filegroup.
You can leave the non-clustered indexes in the old file group (in my personal opinion it has some advantage to have clustered and non clustered index  in different file group especially if the underlying disks are different and write heavy workload)
Remember when you create a non-clustered index by default it goes to the same file group where your clustered index is. If you want non-clustered indexes in a separate file group you need to explicitly mention that. Here is a good example of this.