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Michael Card's hypothesis is interesting, and certainly not outside the realm of possibility. It is quite true that skilled Greeks were enslaved and brought to Italy during times of conquest and revolt, and perhaps even during times of peace between Greece and Rome, such as was the case when Luke was living. But it seems to me that Card's main argument needs evidence to support it. He claims that (1) Luke is a hypocorism based on the name Lucius; and (2) slaves were commonly named according to hypocirisms. Both of these claims are plausible---but plausible isn't good enough. He needs to give some evidence that this was actually the case.
Unfortunately, even then, the case wouldn't be made. Card himself understands this, when he writes: "I admit that this idea is quite a stretch, and my scholarly friends have told me this theory would never wash in the halls of academia." Then he continues, "let us put forth, not as a fact but as a distinct possibility: Luke might have been a slave" (emphasis original). And that's fine---I agree, it is a reasonable hypothesis. But it just doesn't have any good evidence behind it, at present.
Let me also say something about Dick Harfield's answer. He is correct that Luke-Acts is anonymous, and unlikely to have been written by Luke the companion of Paul---although I disagree that this is relevant to your question. But he makes a good point that the tradition about Luke being a physician is not well-established, since it comes from an epistle (Colossians) which is likely to be pseudepigraphical. On the other hand, whether Colossians was written by Paul or not, it is certainly very ancient, and so we have reason to trust it when it affirms Luke as a physician. And I think Harfield has a decent point about Luke being referred to as among his "fellowlabourers." It is, as DJClayworth has pointed out, quite plausible that a slave might be referred to as such. But it seems more likely to me that Paul considered Luke to be closer to his own social standing.
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The question implicitly asks two questions. The first is whether Luke wrote the gospel that now bears his name; the second is whether Luke was a slave.
Luke's Gospel, like all the New Testament gospels was originally anonymous, as was Acts of the Apostles. They were attributed to Luke later in the second century, largely because it was felt that the author must have been a gentile who knew Paul, and only Luke seemed to fit that description. After studying the two works traditionally attributed to Luke, nearly all critical scholars have concluded that he is unlikely to have been the author.
Luke is mentioned in 2 Timothy 4:11 and Colossians 4:14, but both epistles are widely considered pseudepigraphical. In any case, neither reference gives any information that would suggest he was a slave. Colossians 4:14 describes him as a physician, but that does not mean he was a slave. True, the Romans took many Greek physicians as slaves, but they would take any captives as slaves if they had useful skills. Paul was not a Roman and there is no suggestion that Luke was under any kind of restraint from another person. And of course, doubts about the Pauline authorship of Colossians mean that we can not even know whether Luke really was a physician at all.
Philemon is an undisputed epistle of Paul and mentions a Lucas (Luke) at verse 1:24:
Marcus, Aristarchus, Demas, Lucas, my fellowlabourers.
It seems most unlikely that Paul would socially equate a slave with himself by referring to the slave as "my fellowlabourer." On the evidence as I see it, Luke was not a slave.