I think they do. When I checked Maxicare's website for providers, when you toggle down, you can see several Healthway branches. But just to make sure, I suggest you personally check it by visiting this link: https://www2.maxicare.com.ph/Pages/AccreditedProviders.aspx
Yes healthway clinics accept Maxicare cards! :) I've learned it when a rep from Maxicare company itself made an orientation for the company I've worked before! ;)