I've heard a test can go positive with breastmilk during a pregnancy, but that it can also go positive with breastmilk when not pregnant. I believe (but really don't know) that something in breastmilk that is NOT hCG causes a reaction with the antibody strip in the urine test. I have no idea why that is.
In order for hCG to be present in milk to give that solid of a line, it would have to be present in blood in pretty high concentrations. If it's present in the blood in high concentrations, it should be present in urine too, and so you should see a BFP with urine if you are indeed pregnant.