After reading Rex's post about the final quote mark being added automatically, I decided to just leave off that last quote mark, and I found that it works:
echo "foo" | findstr "\" > foofee.txt
Here's why, I believe:
1] As far as TCC is concerned, the findstr command contains an open-quote and a close-quote (because TCC's escape character is ^, not \), and therefore when TCC parses the line it figures that the findstr command ends before the >, and thus it sends findstr a parameter of: "\"
2] As far as FINDSTR is concerned, the backslash *is* an escape character. Therefore, when TCC passes the "\" on to findstr, findstr reads this as: open quote plus literal quote. Fortunately, findstr is just as forgiving as TCC about leaving off the endquote, so it supplements the endquote by itself, and hence it successfully performs the search for a quote mark.