Boone said Rizzo, who turned 32 on Sunday, is experiencing some symptoms but "is doing all right." Rizzo was placed on the COVID-19-related injured list.
Cole tested positive Monday, Montgomery tested positive Tuesday and Sanchez was placed on the COVID-19 list Thursday. Boone said Cole, Montgomery and Sanchez are feeling better.
Players who test positive for COVID-19 are sidelined for a minimum of 10 days before they are allowed to return, per MLB protocols.
The outbreak occurred after the Yankees returned from playing six games against the Tampa Bay Rays and Miami Marlins in Florida, where COVID-19 cases are surging. Rizzo made his debut with the Yankees on July 30 at Miami.
"I think a lot came out of Florida where we were in this wave of them,'' Boone said. "It's a little of you don't know when and where it's spreading.''
At least 85% of New York's players are vaccinated, resulting in a loosening of restrictions, but the Yankees still have been hit hard and Boone said the team will hold conversations about being cautious on its upcoming six-game road trip.
First baseman Luke Voit was activated from the injured list Sunday and replaced Rizzo in the lineup for the series finale against the Seattle Mariners. Voit, who had been sidelined by left knee inflammation, is batting .241 with three homers and 11 RBIs. He led the majors with 22 homers in the pandemic-shortened 2020 season.
"He feels great,'' Boone said. "As you know, he's felt great for about a week now and has been able to get some at-bats. So, he's ready to go and I think excited to be here.''
Rizzo, acquired by the Yankees from the Chicago Cubs before the trade deadline, is hitting .281 with three home runs and six RBIs in nine games since joining New York.
Rizzo, a cancer survivor, said in June while with the Cubs that he wanted to see more "data" before deciding whether to get vaccinated against COVID-19. Boone did not want to speculate on any player's vaccination status.
"A large percentage of us obviously are, but I'm not going to get into that anymore,'' Boone said.
In addition to the four most recent positive tests, outfielder Aaron Judge, third baseman Gio Urshela, catcher Kyle Higashioka and relievers Wandy Peralta and Jonathan Loaisiga also have tested positive for COVID-19 since the All-Star break.
The Associated Press contributed to this report.