It's a rhyme that has connotations from a previous age, so realistically it is bad judgement to use it on tv. I don't believe that Clarkson has said the 'N' word, but I believe he, and the producers of TG, are equally at fault of bad judgement - like it or not, we live in a very PC age and anyone in the public eye needs to be careful not to cause offence - although causing offence is central to Clarkson's appeal and he always sails close to the edge.
For example, there is an instance of this particular rhyme being used during boarding on a Southwest Airlines flight, where a flight attendant encouraged passengers to sit down so the plane could take off using "Eeny meeny miny mo, Please sit down it's time to go". The passengers in question included some African-American's who stated that they were humiliated due to what they called the "racist history" of the rhyme. A jury in the 2003 lawsuit returned a verdict in favor of Southwest and the plaintiffs' appeal was denied.
Clarkson shouldn't (and won't) be sacked, he is worth too much to the BBC and has not been proven guilty at all. Bad judgement, yes, racist slur, no. And no amount of rabble rousing or lynch mobs should be allowed to prevail.