As the title asks, is it genetically possible for a Squib to produce magical offspring, or would the Squib gene trump the magic gene in all cases? How would it be possible for a Squib to have magical children? Additionally, must a Squib come from a pureblooded union, or does the possibility of a half-blood Squib exist?
Yes, it's possible. In fact, it's where "Muggle-borns" come from (though usually after a few more generations).
Muggle-borns will have a witch or wizard somewhere on their family tree, in some cases many, many generations back. The gene re-surfaces in some unexpected places.
I would think so.
There's not much known about the genetics of magical ability.
In my answer to a related question, I pointed out that there isn't a single 'magical gene' that is off or on - if magic power is genetic there must be a number of related genes that control it.
I would think it is possible for a Squib to produce magical offspring, though it would be less likely with Muggles than with other magical beings.
It's likely that squibs result from recessive genes, and are the result of parents who both have the recessive genes (either genes which have a mutation that 'turns off' magical ability, or genes with very weak magical ability). In either of the common cases where recessive genes express a negative effect, random mutation DOES have the ability to reverse this in successive generations (though the odds are very low). More importantly, if it is driven by recessive genes then you can guarantee a child that does NOT express the recessive traits by pairing with someone who lacks the recessive genes.
That being said, I expect 'pureblood' families tend to produce very few squibs, if any at all. Their lack of genetic diversity, however, is certainly beginning to show by the time of Harry Potter, and those detriments certainly outweigh the benefits.