Yes you'd be eligible for Admirals club access in Boston (BA lounge is in Terminal E which is a different terminal to where AA depart from). Just show your BA silver card on entry and you'll receive access. In addition you'll also be eligible for access at your transit city as I'm fairly certain there isn't a nonstop BOS-CUN service with AA - so be that MIA, DFW, CLT, JFK or PHL, you'll have access there too between connections. You're eligible for lounge access on the return from Cancun under caveat of your BA silver card however AA doesn't have an Admirals Club at Cancun in their departure terminal. There's a pretty nasty third party lounge which doesn't allow free access with a BA Silver or when flying AA Business. I believe it's Priority Pass or pay on entry only. My advice is don't waste your cash. The BA flight departs from a different terminal so lounge provisions to their passengers wouldn't be accessible to you flying AA either - and it's another third party lounge hole. All things considered you'll be better off in the main terminal. Again at your US transit city enroute back to BOS you'd be eligible for Admirals Club access upon showing your BA Silver. Under One world rules, you're eligible to access AA lounges with a +1 booked/ticketed on a departing One world flight so if you're flying with a companion then you can guest them in too.
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