“It will be helpful to note that even though neither and either are singular, in informal English they can sometimes take on a plural verb when they are followed by prepositional phrases beginning with of. (In more formal settings they will retain their singular status regardless of the prepositional phrase.)
Neither of the twins are inclined to give thought to what the people outside of their bubble are thinking.
The correct version is “neither of the twins is allergic to penicillin.” When neither or either are used by themselves in a sentence, they take the singular.
However, when neither-nor or either-or is used, the number of the verb is determined by the second subject. For example,
Neither the coach nor the players have arrived.
Neither the players nor the coach has arrived.”