I understand that ink in fountain pen cartridges is a bit thinner. Can I still use it for dip pens if I squeeze it out of the cartridges? ...
First off, a syringe typically works best for extracting or injecting inks into cartridges than squeezing. It's extraordinarily easy to buy blunt-needled syringes these days (walmart.com lists a set of four for <$3) because they're used by folks who want to refill their inkjet printer cartridges.
Fountain pen inks are always water-soluble dyes. This is to make sure that you can always unclog a fountain pen by simply soaking or rinsing with water. The vast majority of fountain pen inks actually use the same commercial dyes that are used on cloth—I think only a handful of brands mix their own (J. Herbin, Noodlers, Private Reserve). So, you could probably just leave the ink uncovered for a while to get some of the water to evaporate which will concentrate the dye. But it won't get any more viscous or less transparent, because it's water based.
However, some fountain pen inks (such as Private Reserve) are known to be harbingers of fungus. So if the ink that came out of the cartridge was sludgy or gel-like, rather than liquid, dump it. It's more trouble than it's worth at that point.
Dip pen nibs, unlike fountain pens, have no hard-to-reach areas, and being metal are safe to use with solvents, so opaque pigment-based inks like india ink can be used with them.