Which free agent starting pitchers have refused to sign in Boston? Price signed here, and he was the premier pitching FA that offseason. Lackey signed here as well, and he was a premium free agent that year. Lester turned Boston down, but there was some history with both Boston and Theo that no doubt influenced his decision. Off the top of my head, there's Mike Mussina, but that's many moons ago.
It's hard to know for certain. Because the Sox were able to acquire quality starting pitching via trade, they were not often bidding for the top free agent pitchers other than those noted above. And they've had good luck keeping their prospective free agent pitchers (Price opted in, Eovaldi resigned, Sale extended, Beckett and Schilling extended their stays here).