Negative value on linear gene expression in microarrays
I'm not personally familiar with Illumina arrays, but I can give some notes here.  This link is a paper which describes the array quality controls specifically. This presentation describes the calculation of the intensities in bioconductor. 

The answer is yes: you will find negative numbers sometimes.  They should be rare.  The Intensities numbers from a scanner which essentially takes an image of the bead fluorescence on the glass slide and tries to subtract a background signal.  

The background is the typical level of signal you see on a bead with no sample DNA bound to it.  You will get some pixels lighting up a bit even there.  

In expression microarrays, this is not a perfect system as each bead has a different nucleotide sequence on it.  Each bead has a specific DNA sequence has non-specific binding which is pretty much unique on the slide.  That is to say, the oligomer on the bead might bind strongly to DNA from your sample which is not its reverse complement to varying degrees.  

I think what happened is that there will be a handful of cases you can find a bead which has less fluorescence than the background controls on the slide.  It's possible that the probe has a design flaw but I would generally assume that the negative number means that there is no detectable target cDNA in the sample for that oligo.  

I probably wouldn't convert the number to a zero, but you could probably justify it to yourself in some cases.  Most difference experiments are the logarithm of the ratio so zero is not a great number in those cases.

@Luke 's comments are well said.  

I think that negative values can still represent signal because of the variance of sequence-dependent effects for the purposes of scaling, and difference experiments.  But I'd tend to think of them as zero signal in any case.  There are probably genes which express at or below detection threshold all the time, which is not zero information, so the number has some value.