Conner, 26, spent parts of the past three seasons with the New York Jets, who waived the former fifth-round pick last October.
The Giants hope the addition of Conner can boost a running game that ranks last in the league, averaging 44.3 yards per game.
"I'm definitely gonna give my best game to help this run game out, cause that's what they brought me in for. So I'm gonna do that to the best of my ability," Conner said.
Conner signed with the Cincinnati Bengals late last season but was cut prior to the start of this season.
The 245-pound Conner will replace Hynoski, who suffered a fractured left shoulder in Sunday's loss to the Carolina Panthers.
"We need a fullback, we need a leading blocker," Giants coach Tom Coughlin said. "[Conner] is a physical guy -- we think he'll go up in the hole and clear the hole for the running back, and we hope he'll be a good pass protector and contribute on special teams as well."
Conner was a favorite of Jets coach Rex Ryan and became a reality TV sensation during the team's 2010 appearance on HBO's "Hard Knocks."
Nicknamed "The Terminator" after the character of the same name in the "Terminator" movies, Conner was praised by Ryan for his blocking ability.
ESPN NFL Insider Adam Schefter, ESPN.com Giants reporter Dan Graziano, ESPNNewYork.com's Kieran Darcy and ESPN.com Jets reporter Rich Cimini contributed to this report.