I have a plant whose soil is infected with 1/4 inch worm or caterpillar type things. They come out the bottom of the pot and die. Now I am finding the dead ones through out the room where the plant is located. The plant is not especially happy but not dying either. The things have been there for at least a year and I tried using an insecticide in the soil. Obviously that didn't help. What are these things? Should I just dig the plant out, wash all the soil off the roots and repot in new soil? I am afraid they will spread to my other houseplants.

If these critters are grayish or brownish in color with many legs and look like they are wearing armor, what you might be seeing are called "Millipedes" If touched, they curl up. Don't worry, they will not sting or bite you. They grow between 1/4 to an inch in length. They need to be in a moist atmosphere or they will dry up as you have seen on your carpet.
You can try flooding the pot outdoors with lots of water, where as, they should come to the surface trying to escape the flood. Another way is removing the old soil and replacing it with new, outside of course, before they have up to 300 more larvae/babies! It has unfortunately happened to me!