What are the lease terms? Your lease should specify how much notice you need to provide that you are not renewing your lease. If it does not specify, then state law will control, which probably requires much less than 60 days. (In my state, for example, only 15 days.) Do your research.
Obviously, if you gave the proper notice required by the lease or the law, your landlord is in the wrong. But let s say you didn t give enough notice. In that case, you might be on the hook for the remaining time--until June 7, your landlord said. However, once you moved out, your landlord had a duty to try and find new tenants as soon as possible--and once the new tenants moved in, you could not be made to pay rent for any days afterward. If it occurred in the middle of the month, then you would only have to pay for the days it stood empty, as SuperTramp explains.
You have one last out: what does your lease say about oral agreements? Unless it says that the lease can only be modified in writing, the landlord might have been barred from claiming any more rent by telling you on May 31 that you had no further obligations. This would be harder to prove in court than something solidly contained in the lease, however.