Recent approaches to word sense disambiguation (WSD) utilize encodings of the sense gloss (definition), in addition to the input context, to improve performance. In this work we demonstrate that this approach can be adapted for use in multiword expression (MWE) identification by training models which use gloss and context information to filter MWE candidates produced by a rule-based extraction pipeline. Our approach substantially improves precision, outperforming the state-of-the-art in MWE identification on the DiMSUM dataset by up to 1.9 F1 points and achieving competitive results on the PARSEME 1.1 English dataset. Our models also retain most of their WSD performance, showing that a single model can be used for both tasks. Finally, building on similar approaches using Bi-encoders for WSD, we introduce a novel Poly-encoder architecture which improves MWE identification performance.