I think there is no straight forward option in BLAT to collect unmapped reads. May be you can try this.

1. Collect your mapped reads using cut command first.

   cut -d "       " -f 10 output.psl | sort -u >mapped_header.txt ## 10 here is my mapped read column.
   

2. Now collect all your all read headers.

   LC_ALL=C fgrep ':N:' sample.fastq >all_header.txt 
   

   I used this pattern ":N:" because it is present in all my headers. If your read also has similar pattern you can probably use this or use something common in all the reads like "@HISEQ" or "@HWI" etc

3. Now collect those headers which are unmapped using following command.

   awk 'NR==FNR{a[$0];next}!($0 in a)' mapped_header.txt all_header.txt >unmapped.txt
   

4. Now grep those reads from the original fastq file.

   LC_ALL=C grep -A 3 -F -f unmapped.txt sample.fastq >unmapped.fastq
   

Since, we are searching for fixed strings, the LC_ALL grep would not take too much time.