first (trivial) answer: the spectrum of a bipartite graph is symmetric wrt to 0; hence, +1 is an eigenvalue iff -1 is an eigenvalue.
second (trivial) answer: an individual edge has eigenvalue +1 (and hence also -1).
further (much less trivial) answers: take a look at this article by chan and godsil, where several conditions answering your questions are presented, e.g. at page 76. in a slightly different version you can find on the internet (google is your friend), the same authors show that -1 is an eigenvalue if the graph has a perfect 1-code. also, in this monograph you can find some relevant information, e.g. the observation that +1 is an eigenvalue of so-called collinearity graphs.