This is a practical question. I’m an options trader and wanted to hire somebody to help me during the yom tovim. This individual tells me that he is catholic. Then, one day, he tells me that his father told him when he was 17 that his mother, from Hungary, was Jewish. What is the halacha? In a perfect world I wouldn’t need him but he is the best qualified to help me. Is he believed? Does somebody who tells you that that they are not Jewish but that their mother is, is he believed? Can you hire such a person? Can you lend that person money at interest? I’m not interested in knowing the chumra, that’s obvious. What is the actual halachic status of such a person?
The man has a chezkas goy (because he acts as a Christian, he is assumed to be one). Becasue the testimony concerning was given by the individual’s father (who is definitely a non-Jew), it has no bearing with regard to matters of halachah. You may therefore continue to assume he is a non-Jew, unless presented with concrete proof to the contrary — and he may continue to work for you on Yom Tov (I haven’t addressed the issue of having non-Jews working for Jews on Yom Tov).
Concerning the general question, somebody who acts (and has always acted) like a non-Jew, and claims his mother is Jewish, is not believed, and may be treated like a non-Jew for all intents and purposes. However, if proof is produced the case would depend on the halachic validity of the given proof.
Sources: Yevamos 122b; Rashba 1:243; Shach YD 127:20; Yam shel Shlomo Chulin chap. 8, no. 65; Beis Yehuda, YD 1; Shaarei Tzedek (Geonim, sec. 30) [concerning a slave who claims she her mother is Jewish]; Shulchan Aruch Yd 268.