The Costa Rica international will have surgery on Monday after a challenge by Stevenage midfielder Simon Heslop during Saturday's FA Cup clash
Oviedo was carried off the pitch after 22 minutes of the match at the Lamex Stadium following a challenge with Simon Heslop.
After the game the Toffees confirmed, via their Twitter account, the severity of the injury: "Bryan has broken two bones in his left leg. He will remain in hospital over the weekend and have an operation on Monday."
The 23-year-old, who has made 13 appearances for the Merseysiders this season, now look set to miss out on this summer's World Cup, with Costa Rica drawn in the same group as England, Italy and Uruguay.
Everton went on to win the game 4-0 thanks to two goals from Steven Naismith and strikes from John Heitinga and Magaye Gueye.