First, you need to let your achilles tendon heal. This means putting less stress on it. That means resting for a while and getting a shoe with a higher heel and/or more support. Warning: This can take a *long time* :(
Once you're past the acute phase of injury, minimalist shoes could, in theory, offer some benefit by helping to stretch out your calves and related tissues. More limber calves exert less stress on your achilles tendon, meaning you could avoid a relapse. But minimalist running in general puts more stress on your achilles, so...kind of a catch-22. If you go minimalist, go slowly and carefully.
I tried to do this with my achilles problems, but I think my calves are just too...short. Concerted stretching and strengthening campaigns have helped, but I still run into problems if I try to run in something with too low a heel drop.
With achilles issues perhaps more than others, YMMV.
Some people's bodies can handle it, and some can't. Whichever camp you're in, proceed with caution.