Whitehurst completed 14 of 24 passes for 182 yards and a touchdown before injuring his knee in the fourth quarter of Sunday's 33-13 loss to the New England Patriots. The 34-year-old entered the contest after rookie Cody Kessler sustained rib and chest injuries in the first quarter.
Cleveland reached an injury settlement with Whitehurst before releasing him.
McCown, who sustained a broken collarbone in Week 2, is expected to return to practice this week as Cleveland (0-5) vies for its first win on Sunday (1 p.m., ET) when it visits the Tennessee Titans (2-3).
Hogan was selected by Kansas City in the fifth round of the 2016 draft before being released by the Chiefs in September. The 23-year-old, who was signed to the Browns' practice squad shortly thereafter, completed 727 of 1,103 passes for 9,385 yards with 75 touchdowns and 29 interceptions in his collegiate career with Stanford.
Cleveland also signed defensive lineman Gabe Wright to the active roster and inked wide receiver Mitch Mathews to the practice squad.
Wright, 24, recorded one tackle in two games this season after collecting six in seven contests with the Detroit Lions last season.
Mathews was signed by Kansas City as an undrafted free agent in the summer before being waived on Aug. 30. The 6-foot-5, 220-pound rookie recorded 152 receptions for 2,083 yards and 24 touchdowns in 41 career games at BYU.