Over the past few decades, the finite element method (FEM) has established itself as a powerful numerical technique for solving various electromagnetic problems.  Despite great progress in recent years, the FEM has commonly recognized issues such as the laborious and time-consuming task of meshing three-dimensional bodies of complex shapes.  Compared to the FEM, the meshing tasks are generally simpler for the boundary element method (BEM) that requires meshing of the bounding surface of the interest domain only.  However, even with the dimensionality advantage over the FEM, the BEM still demands significant tasks of meshing or re-meshing of the boundary surfaces for industrial applications, such as eddy current nondestructive inspection simulations, involving complex geometry components and deformations.  For such problems, development of advanced methods with simpler meshing requirements is desirable.  Various domain-based and boundary-based meshless methods have been proposed to simplify meshing tasks of the FEM and BEM in the field of computational mechanics.  The main idea of the meshless methods is to make approximations entirely in terms of distributed nodes rather than using elements.  This paper presents a meshless boundary integral equation method for solving eddy current problems. Using the moving least squares approximation, the proposed method adaptively discretizes the boundary integral equations for electromagnetic fields, and hence retains the meshless attribute of the moving least squares approximation and the dimensionality advantage of the BEM.  By solving a two-dimensional eddy current problem for non-trivial boundary geometries, we explicitly demonstrate these features, namely that the method has the potential to achieve better accuracy than the conventional BEM, while alleviating meshing tasks and retaining the dimensionality advantage of the BEM over the FEM.  It is anticipated that similar computational advantages and accuracies will manifest themselves for three-dimensional electromagnetic problems, including both electromagnetic wave propagation and eddy current phenomena.