I guess my memory of Nasar's book's contents was slightly incomplete. I found a few more sources by googling:
Ed Regis "Who Got Einstein's Office" (a history of IAS) mentions the incident briefly as part of a longer discussion about Milnor. It doesn't say anything about homework.
This is probably the best one: http://dspace.sunyconnect.suny.edu/bitstream/1951/37717/1/Stony%20Brook%20Press%20V.%2012,%20N.%2015.PDF It's a scan of a newspaper from Stony Brook, where page 2 has an article about a symposium commemorating Milnor's 60th birthday. It quotes Tucker himself telling the story about Milnor's solution. It sounds like Tucker described the conjecture in his class as being open, but jokingly suggested that it was a homework assignment. Maybe that's the origin of the "homework" story. It mentions Milnor gave Tucker his solution just 3 days later.
Mathforums.org has someone posting an email from Tucker's son, that differs from the above in a few details: http://mathforum.org/kb/message.jspa?messageID=1186409&tstart=0
It also appears in a Princeton Alumni Weekly story from 1958. This one isn't so great; its main interest is that it comes from that long ago. http://www.princeton.edu/~mudd/finding_aids/mathoral/pmcxpaw.htm#boywonders
This talk (from his home page) doesn't mention the incident at all, but has a few unrelated anecdotes in the Q&A section: http://www.math.sunysb.edu/~jack/PREPRINTS/index.html I mention it because it makes me guess that he probably doesn't mind talking about this type of story.
Someone probably ought to email Milnor just to let him know he's being discussed here (it's a bit impolite for us to be doing this without informing him).