First off, the wizards who call muggle-borns "mudbloods" are not full of wickedness in their minds like they are under some evil spell; They are simply evil hateful racist people bent on taking over and destroying everything good.
A muggle-born is a person with magical abilities born into a family of muggles, hence the name muggle-born: they are just like any other wizard or witch and they are not inferior to, or weaker then pure-blood witches and wizards.
They are referred to as mudbloods in a racist term as pure-bloods consider themselves superior because of their lineage.
A muggle-born is not handicapped in any way; in fact some purebloods may be more hindered then a muggle-born (because to maintain a pureblood family some incest may be required).
A squib is a "muggle" born into a family with one or more magically gifted parents. The books don't mention any other names that Voldemort and his gang use to refer to squibs.
A squib would not be considered a muggle-born because he or she would have to have at least one magically gifted parent and no magical ability to count as a squib where-as a muggle-born has both of his/her parents muggles and magical abilities
A child of a squib would either be a squib or a wizard/witch because of genetics. One of his parents was a squib so it doesn't matter if the child has the other parent as a muggle or a magical, the child still has the genetic potential to have that magical ability.
Just because the parent is a squib does not mean that the genetic trait (or lack of) that allows magic isn't there. It would be passed down to his or her children giving them the ability to have magic. (Or not, it depends on what genetic traits dominate.)
According to this post a squib is not a muggle, and for a muggle-born to occur both parents would need to be muggles. So a child of a squib would either be a half-blood or a pure-blood (depending on the parents).