If I apply for baby’s medical, will my boyfriend have a problem whenever he applies for green card?

A U.S. citizen's application for needs-based public assistance will not stand in the way of eligibility for her husband's eligibility to become a Permanent Resident. Note, however, that there may be very significant problems associated with your fiance's plan to enter the U.S. with a visitor's visa while planning to marry you and then apply to adjust status (i.e., to remain permanently with a "Green Card"). In fact, that may constitute visa fraud with very harsh immigration-related consequences. There really is no substitute for you and your fiance to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options, and strategies to attain your goals, and who then could offer legal representation in the often complex application process.