The duo become the third and fourth players on the Devils to test positive for the disease, joining center Travis Zajac and defenseman Adam Larsson. Zajac is already over the disease and back to skating on his own, a team representative confirmed earlier this month to ESPN.com.
Elias, 38, has four goals and nine assists in 29 games this season. He is expected to skate in the next few days.
Ryder, 34, has four goals and 10 assists in 34 games.
Goalie Scott Clemmensen, who plays for the Devils' AHL affiliate in Albany, also was diagnosed with the mumps.
The league sent out memos in mid-November to all NHL clubs, in coordination with the NHL's infection control subcommittee, to remind teams' medical staffs and head trainers of the recommended protocol and guidelines for diagnosis and prevention.
It is not known where or when this particular outbreak began within the NHL, and it is unclear just how long the problem could persist.
ESPN.com's Katie Strang contributed to this report.