What is a "unit investment trust"?  Can a unit investment trust invest
in non- publicly traded stock?

http://www.sec.gov/answers/uit.htm

& no they cannot invest in non-public companies.

Thank you for your response.  You wouldn't have authority for the
proposition that UITs can't invest in non-public companies, would you?

Unit Investment Trusts are regulated by the Investment Company Act of
1940 (see the 3d link) which defines that they invest in securities. 
The definition of "securities" can be found under no. 36 on the 4th
link.  As I understood it, this only includes shares that are traded
on an exchange, ie, not non-publicly traded stock.

The 1st link is the most extensive discussion of UITs that I found. 
Others agree entirely.

http://www.ici.org/funds/inv/bro_g2uits_p.pdf
http://moneycentral.msn.com/quickref/quickref.asp?Cat=10&SelCat=3&RefType=0&QAMode=2&Topic=8&Sub=5
http://www.law.uc.edu/CCL/InvCoAct/sec3.html
http://www.law.uc.edu/CCL/InvCoAct/sec2.html

I hope this helps you.  If you have any questions, post them as a
"clarification" to your question as that will be more noticeable to a
Researcher (blue name), who can post an official answer.  But, of
course, I will be pleased to try to help.