Are their other ways genes can mutate besides SNPs?

Genetic code can be "misspelled" in many ways besides a single nucleotide polymorphism. These include deletions, copy repeats, gain, loss, or inversion mutations among others. The StrateGene test cannot detect most of these more complicated structural errors due to the limitations of the Infinium Global Screening Array-24 v3.0 BeadChip used.

However, we are able to call three insertion/deletion mutations in the ADORA2A, DHFR, and TYMS genes.
These insertion/deletions are not SNPs, they are short runs of several nucleotides in a row that are either missing entirely (deletion) or have multiple repeats (insertions).  Researchers who specifically want to study these types of mutations use a different technology than the Illumina array to detect them with a high degree of accuracy. 

You can read more here: http://www.ncbi.nlm.nih.gov/dbvar/content/overview/