I could see the Sox - not exactly shopping Bogaerts - but telling teams to make an offer. Now I am not sure how robust the market it is. I tend to chock up this year to injury ... nothing like a hand injury (regardless of what he says to people) to mess up your swing. It is not something which get better during the season. That said, if you save the brinks truck for Betts - then Bogaerts and Bradley are the swing guys here.
Sox need another bat - but the trade market is probably where the best possibility is to get one ... and given that the Sox have used a lot of their best minor league pieces already, looking at one of their young guys has to be on the table. I am sure there will be chatter about trying to get Stanton - which I think for a bunch of reasons (payroll flexibility, cost in players, marginal value) is stupid. I could see kicking the tires on Andrew McCutchen - though again, you only get a year of him and I am not sure what sort of match there is with the Pirates given that.
The free agent class looks horrid. That said, if last year's market is a barometer for the future - it could be a buyer's market for slow, power hitting non-defenders ... the sort of thing where a JD Martinez might be gettable for a surprisingly short hitch a la Encarnacion last year. Getting something like that makes sense. Same with Jay Bruce (less appetizing option) for a 1-year.