The "X-one to one" report returns a list of matches in which you share X-DNA. Women receive 2 copies of X-DNA, one from their mother and one from their father. Men receive only one copy from their mothers. The X-chromosome is often referred to as the sex chromosome. XY for males (hence why men get 1 copy) and XX (for females).
X-DNA unlike autosomal DNA actually follows a pattern of inheritance. It can recombine just like autosomal DNA (meaning you may or may not receive all or some segments passed down by your parents), however X-DNA is passed down from a female. So the MRCA (most recent common ancestor) for you and your matches will be a female ancestor.
In your case you may not have any X-DNA matches as of yet if you used the X One-to-Many report. As more people upload their results to Gedmatch, you'll start to see more matches appear. If you ran the One-to-Many matches report, then you have the option to access the X-DNA details between you and the individual. You must however make sure that you are looking at the correct column that indicates that you and the match actually share X-DNA. When you viewed the data for this cousin, did Gedmatch indicate that you shared X-DNA or autosomal DNA only? I'm assuming that you ran the One-to-Many matches report that it generated a list of matches and you happened to click on the X-DNA link, which returned nothing. If this is the case, then you will need to look in the column to see if Gedmatch has determined whether you share X-DNA. If you used the One-to-One matches tool and you selected X-DNA to be analyzed and a cousin appeared as an X match, but when you wanted further analysis it appeared as no match, then you do have a problem and I'd reach out to the developers of Gedmatch about the situation. It could be a software glitch.
Just to help you understand your X-DNA a little more, I would suggest you read the DNA Explained blog for clarification. I've posted a link below that gives an explanation about X-DNA. The blogger has also written other articles about the X-chromosome that you may want to read as well. I would suggest you print off a copy of the X-DNA chart so that you can fill it out and see which ancestors contributed X-DNA. Most of the charts look like a fan, which I'm not crazy about. There are other charts that have a different look that contains the same info.
http://dna-explained.com/2014/01/23/that-unruly-x-chromosome...I wish you luck in your journey. Also you may want to join some of the social media groups on Facebook. There are a few yahoo and Google groups as well. You'll find super helpful and knowledge people in any of these groups.
LaKeisha