Who would have guessed that there's a forum discussion about how to refill the container for the Swiffer solution? Are there really little groups of people out there discussing things like this? Yes, thank goodness. Anyway, in the past, I tried to punch a hole in the top of my Swiffer bottle, refill it and attach a little piece of duct tape over and over again. It never worked. So, here's the collective solution to refilling the solution. You can take the cap off the bottle by soaking it in very hot water. You may need to use pliers to get the cap off. Now, refill it with your own cleaning solution. The beauty of this is that you can use 2 parts vinegar and 1 part water, a dilution of dishwashing soap and a disinfectant, or, why not dilute Murphy's oil soap and use it for the wood floor cleaner.
Also, there seems to be a myth that the Swiffer floor cleaners are poisonous or harmful to pets. Again, the forums discuss this in depth and there seems to be no evidence that pets are harmed from the Swiffer products.