I was wondering if GATK uses the MAPQ column in the sam file in any of its calculations?
What aligner are you using to map your reads? If the aligner doesn't emit MAPQs, then yes you need to use the filter to assign them arbitrarily. By doing so, you lose some power for the variant calling, but you have no choice, unless you can use a different aligner that does assign MAPQs.
Several GATK tools (e.g. the variant callers) use MAPQ, yes.
I see, but then how will the following tool affect variant calling? Also, would it be incorrect to randomly assign some arbitrary value for MAPQ in the sam file (since Illumina doesn't output anything for that attribute)?