I had an employee who quit my center for another job.  Her new job
fell through and she filed for unemployment.  The state says I'm
eligable to pay her unemployment.  However, if I say I would rehire
her - am I still eligable?? (Despite her decision to come back or not
)

Thank you very much for accepting my comment as the answer to your
question. I am reposting it, with an additional item that I hope will
be helpful.

"Generally, voluntary terminations, refusal to accept a reasonable
offer of employment or reemployment, unavailability for work, and
misconduct may be disqualifying conditions under New York State
Unemployment Insurance Law."
 
University of Rochester
http://www.rochester.edu/admin/HR/policies/273.htm

It's heavy reading, but you may want to look through the New York
State Department of Labor's pages of information on Unemployment
Insurance:

New York State Department of Labor
http://www.labor.state.ny.us/business_ny/unemployment_insurance/unemployment_insurance.html

Of particular interest may be this section, on "refusal of employment"
by the claimant:

New York State Department of Labor
http://www.labor.state.ny.us/business_ny/unemployment_insurance/interp/Section_1200.htm

Google search strategy:

Google Web Search: "new york" "unemployment insurance" + "refusal" +
"reemployment"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22new+york%22+%22unemployment+insurance%22+refusal+reemployment

Please request clarification if needed; I'll be glad to offer further
assistance before you rate my answer.

Best wishes,
pinkfreud

"Generally, voluntary terminations, refusal to accept a reasonable
offer of employment or reemployment, unavailability for work, and
misconduct may be disqualifying conditions under New York State
Unemployment Insurance Law."

http://www.rochester.edu/admin/HR/policies/273.htm

thanks pink - a useful link - it pretty much answered my question. 
You also posted a comment to another one of my questions recently that
was quite helpful.  So please definitely post this as the answer.  I
appreciate your high level of standards as to what you consider to be
an accurate and quality answer.