Also, if you feel comfortable telling us in which state you reside, I'm certain someone can quote you the specific laws and guidelines covering polygraph testing.

Polly-Anna Poly-Ann,

What the judge is doing may well be legal in your state. I suggest you consult with a lawyer on this. Whether it is legal or not, the judge is either ignorant, incompetent, or both to have suggested that the matter be settled by a pseudoscientific polygraph "test." You were right not to agree to this, and while I am not competent to dispense legal advice, it seems to me that you might have sufficient legal ground to have this judge removed from the case.

Your erstwhile tenant could well beat the "test." (See Chapter 4 of The Lie Behind the Lie Detector to find out how.) That polygraph "tests" are easily beaten through the use of simple countermeasures that polygraphers cannot detect (despite their unsupported claims to the contrary) is reason enough for placing no reliance on them.

There is not all that much difference between the kind of polygraph "test" that would likely be used in your situation and those commonly used in pre-employment screening. Both would likely be probable-lie "Control" Question "Tests." See Chapter 3 of The Lie Behind the Lie Detector for further details.

In explaining your refusal to allow your case to be settled by a pseudoscientific trial by ordeal, feel free to refer the judge to The Lie Behind the Lie Detector (you might want to give him a printout) and to AntiPolygraph.org.

I wonder if anyone has any experience or advice about the use of polygraphs in a non-criminal court case.  I have a situation which seems impossibly strange and I don't know what to think... or do.

The background: I had to go to court to evict a tenant, who was ordered by the judge to vacate and return certain expensive personal property to me and also to refrain from vandalizing the house.  The individual did not return the property and did thousands of dollars of damage to the house, so I went back to the judge, asking that the individual be held in contempt of court.  The judge listened to my testimony and my witnesses and then the defendant's denials. Before listening to any of the defendant's witnesses, the judge interrupted and asked the defendant if he would let the matter be settled by a polygraph administered by a State Police Examiner . The defendant heartily agreed and my lawyer suggested that we go along (which I now regret though if I had objected it would probably have upset the judge). As we were adjourning, the judge asked me, almost as an afterthought, if I would also agree to take a polygraph.  I declined, saying that I had nothing to prove, but the judge gave me a very skeptical look. Please consider that this is taking place in a very small town in a somewhat rural area of the South.

Questions: 
--Can this be legal?  
--Does anyone have any similar experience?
--How might State Police examinations differ from job related polygraphs?  (I'd prefer not to reveal the particular state publicly.)
--What if he "beats" the polygraph, appearing innocent of the theft and vandalism?  He made the counterclaim that I fabricated the charges for spite, which the judge might believe if I don't also take the exam. 

Any thoughts or advice would be gratefully received!