A: Tu n'as pas confiance en nous?
B: Non... C'est même l'inverse que je redoute. Aussi préféré-je garder certaines choses secrètes.
I’m having trouble working out what exactly these two phrases mean in this context.
"Même" is used for emphasis in that context, similar to the emphatic use of "even". It can also mean "same" in other contexts ("c'est la même chose") but here it's clearly just emphasis, so you can basically ignore it when figuring out the meaning. And "l'inverse" is "the opposite", so:
B: No... it's quite the opposite i'm afraid of.
"Opposite" of what? Presumably B is afraid of A being too trustworthy? Context should clarify things.
"Aussi" means "also" or "hence" (clearly the latter here). As for "préféré-je": in some cases, the subject and verb are inverted in affirmative sentences. A sentence beginning with "aussi" is one of those cases, along with "peut-être" and sometimes a place ("sur le mur était perché un oiseau"). It's a rather formal / scholarly phrasing used mainly for style and emphasis. Thus, "je préfère" becomes "préfère-je", which becomes "préféré-je" for euphonic reasons. The second part translates to:
Hence, I prefer to keep some things secret.
I think a translation is the simpler way to explain it. "préféré-je" is a formal interogative present form.
A : Don't you trust us ?
B : No... It's quite the contrary that I fear. Thus I prefer to keep certain things secret.