Equality of single bits is the XNOR function. There is no unconditionally secure protocol for XNOR in the presence of malicious adversaries. We prove this in:
HK Maji, M Prabhakaran, M Rosulek: Complexity of Multiparty Computation Problems: The Case of 2-Party Symmetric Secure Function Evaluation, TCC 2009.
There is a simple protocol secure against semi-honest adversaries, but keep in mind that if you know your input bit, you know whether your bit is equal to the other guy's bit, then you know the other guy's bit. So the secure protocol is "just tell the other guy your input."
On the other hand, equality of strings 2 bits or longer does not even have a protocol secure against semi-honest adversaries. This follows from an old characterization of Kushilevitz: the function is not "decomposable".
E Kushilevitz: Privacy and Communication Complexity. FOCS 1989
You can get equality test if you assume an ideal oblivious transfer. This is not unreasonable since you can get oblivious transfer from physical assumptions (like noisy channels). The private equality test described in this paper is pretty straight-forward, and unconditionally secure against semi-honest adversaries assuming an ideal oblivious transfer:
B Pinkas, T Schneider, M Zohner: Faster Private Set Intersection Based
on OT Extension, Usenix 2014