The question posed doesn’t actually make sense. Non religious Jews would by definition be Jewish.
When I approach someone on the street, or wherever, and ask if he’d like to put on Tefilin, first I ask: “Excuse me, are you Jewish by any chance?” Most people say no. Most Jews ignore me. Of the Jews that don’t ignore me, most don’t want to put on Tefilin. When someone says they’d like to put on Tefilin, and they say they’re Jewish, there’s no reason to assume otherwise. It’s not Siddur Kiddushin. Occasionally, I will ask someone which of their parents was Jewish, as not to give away the “correct” answer.
When we get someone involved in our Chabad Houses and communities, we almost always know whether they’re legitimately Jewish or not. Their Bubby’s and Zaides were often Frum or traditional. When admitting a child into our Hebrew School, Chabad HQ (Merkos) policy is that they must be verified as Jewish. When being Mesader Kiddushin we do a proper Birur Yahadus, often through a competent Rov or Bais Din.
In short, Haimy, you don’t know much about Chabad or non religious Jews, so please educate yourself before being Motzi Shem Ra on others.